L(s) = 1 | − 8·2-s − 12·3-s + 40·4-s + 96·6-s − 21·7-s − 160·8-s + 90·9-s + 33·11-s − 480·12-s + 52·13-s + 168·14-s + 560·16-s − 37·17-s − 720·18-s + 97·19-s + 252·21-s − 264·22-s − 40·23-s + 1.92e3·24-s − 416·26-s − 540·27-s − 840·28-s + 292·29-s + 257·31-s − 1.79e3·32-s − 396·33-s + 296·34-s + ⋯ |
L(s) = 1 | − 2.82·2-s − 2.30·3-s + 5·4-s + 6.53·6-s − 1.13·7-s − 7.07·8-s + 10/3·9-s + 0.904·11-s − 11.5·12-s + 1.10·13-s + 3.20·14-s + 35/4·16-s − 0.527·17-s − 9.42·18-s + 1.17·19-s + 2.61·21-s − 2.55·22-s − 0.362·23-s + 16.3·24-s − 3.13·26-s − 3.84·27-s − 5.66·28-s + 1.86·29-s + 1.48·31-s − 9.89·32-s − 2.08·33-s + 1.49·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.1779979604\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1779979604\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | $C_1$ | \( ( 1 + p T )^{4} \) |
| 3 | $C_1$ | \( ( 1 + p T )^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - p T )^{4} \) |
good | 7 | $C_2 \wr S_4$ | \( 1 + 3 p T + 493 T^{2} - 248 T^{3} + 41604 T^{4} - 248 p^{3} T^{5} + 493 p^{6} T^{6} + 3 p^{10} T^{7} + p^{12} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 3 p T + 3899 T^{2} - 131598 T^{3} + 6860532 T^{4} - 131598 p^{3} T^{5} + 3899 p^{6} T^{6} - 3 p^{10} T^{7} + p^{12} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 37 T + 4895 T^{2} + 342966 T^{3} + 28554948 T^{4} + 342966 p^{3} T^{5} + 4895 p^{6} T^{6} + 37 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 97 T + 6430 T^{2} - 680153 T^{3} + 101426702 T^{4} - 680153 p^{3} T^{5} + 6430 p^{6} T^{6} - 97 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 40 T + 29660 T^{2} - 161700 T^{3} + 397504662 T^{4} - 161700 p^{3} T^{5} + 29660 p^{6} T^{6} + 40 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 292 T + 78578 T^{2} - 13583712 T^{3} + 2593460619 T^{4} - 13583712 p^{3} T^{5} + 78578 p^{6} T^{6} - 292 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 257 T + 100375 T^{2} - 21050774 T^{3} + 4264549156 T^{4} - 21050774 p^{3} T^{5} + 100375 p^{6} T^{6} - 257 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 3 T + 108382 T^{2} + 13583767 T^{3} + 5297078022 T^{4} + 13583767 p^{3} T^{5} + 108382 p^{6} T^{6} - 3 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 385 T + 42500 T^{2} + 4512237 T^{3} - 211934742 T^{4} + 4512237 p^{3} T^{5} + 42500 p^{6} T^{6} - 385 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 654 T + 330400 T^{2} + 100183978 T^{3} + 31726169478 T^{4} + 100183978 p^{3} T^{5} + 330400 p^{6} T^{6} + 654 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 676 T + 268208 T^{2} + 92565930 T^{3} + 33807666231 T^{4} + 92565930 p^{3} T^{5} + 268208 p^{6} T^{6} + 676 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 108 T + 411194 T^{2} + 44802972 T^{3} + 79177712223 T^{4} + 44802972 p^{3} T^{5} + 411194 p^{6} T^{6} + 108 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 313 T + 823859 T^{2} + 186184344 T^{3} + 253544518890 T^{4} + 186184344 p^{3} T^{5} + 823859 p^{6} T^{6} + 313 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 511 T + 255811 T^{2} - 49523318 T^{3} + 57088436528 T^{4} - 49523318 p^{3} T^{5} + 255811 p^{6} T^{6} - 511 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 170 T + 906100 T^{2} + 57984670 T^{3} + 357702187259 T^{4} + 57984670 p^{3} T^{5} + 906100 p^{6} T^{6} + 170 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 303 T + 1293734 T^{2} - 282692763 T^{3} + 669609815142 T^{4} - 282692763 p^{3} T^{5} + 1293734 p^{6} T^{6} - 303 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 602 T + 309712 T^{2} - 1287566 T^{3} + 97146551726 T^{4} - 1287566 p^{3} T^{5} + 309712 p^{6} T^{6} + 602 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 709 T + 1261324 T^{2} - 614954417 T^{3} + 805422854882 T^{4} - 614954417 p^{3} T^{5} + 1261324 p^{6} T^{6} - 709 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 1647 T + 1995455 T^{2} + 1537586100 T^{3} + 1235374382334 T^{4} + 1537586100 p^{3} T^{5} + 1995455 p^{6} T^{6} + 1647 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 490 T + 2710388 T^{2} - 1003810110 T^{3} + 2829012343014 T^{4} - 1003810110 p^{3} T^{5} + 2710388 p^{6} T^{6} - 490 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 70 T + 2842288 T^{2} - 201876122 T^{3} + 3637473889790 T^{4} - 201876122 p^{3} T^{5} + 2842288 p^{6} T^{6} - 70 p^{9} T^{7} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.46434148941494018849591164742, −5.90904218002164944678716650961, −5.90680429626477187975158678743, −5.84708310333761370749485649610, −5.71520521678097440722631381940, −4.92615826305236566506504416829, −4.86997856204751835378141510435, −4.84806954212754639808736356674, −4.82219974712772292634449355173, −4.06768253957149148630293176734, −3.87586877641851879508791400872, −3.75565094334335443314093394431, −3.54279954074923665616732411191, −3.08391913684303714611866094977, −2.78345511687121927004787381999, −2.71323608948355218643349865024, −2.55635941352972983275407351551, −1.70595943765536608595191381585, −1.61448394248340609459645907016, −1.57579166295792913718044330304, −1.31813556159510335419752120063, −0.76626298688517093163898541551, −0.73838592019468602862423291412, −0.51577097178569759151083760304, −0.14084058121355541568562903266,
0.14084058121355541568562903266, 0.51577097178569759151083760304, 0.73838592019468602862423291412, 0.76626298688517093163898541551, 1.31813556159510335419752120063, 1.57579166295792913718044330304, 1.61448394248340609459645907016, 1.70595943765536608595191381585, 2.55635941352972983275407351551, 2.71323608948355218643349865024, 2.78345511687121927004787381999, 3.08391913684303714611866094977, 3.54279954074923665616732411191, 3.75565094334335443314093394431, 3.87586877641851879508791400872, 4.06768253957149148630293176734, 4.82219974712772292634449355173, 4.84806954212754639808736356674, 4.86997856204751835378141510435, 4.92615826305236566506504416829, 5.71520521678097440722631381940, 5.84708310333761370749485649610, 5.90680429626477187975158678743, 5.90904218002164944678716650961, 6.46434148941494018849591164742