| L(s) = 1 | + 2·2-s + 4-s + 4·7-s − 2·8-s − 2·11-s + 8·14-s − 4·16-s − 4·17-s − 12·19-s − 4·22-s − 4·23-s + 4·28-s − 12·29-s − 8·31-s − 2·32-s − 8·34-s − 32·37-s − 24·38-s − 2·41-s + 10·43-s − 2·44-s − 8·46-s + 4·47-s + 12·49-s + 24·53-s − 8·56-s − 24·58-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s + 1.51·7-s − 0.707·8-s − 0.603·11-s + 2.13·14-s − 16-s − 0.970·17-s − 2.75·19-s − 0.852·22-s − 0.834·23-s + 0.755·28-s − 2.22·29-s − 1.43·31-s − 0.353·32-s − 1.37·34-s − 5.26·37-s − 3.89·38-s − 0.312·41-s + 1.52·43-s − 0.301·44-s − 1.17·46-s + 0.583·47-s + 12/7·49-s + 3.29·53-s − 1.06·56-s − 3.15·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5094239342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5094239342\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 4 T + 4 T^{2} + 8 T^{3} - 17 T^{4} + 8 p T^{5} + 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 2 T - 13 T^{2} - 10 T^{3} + 124 T^{4} - 10 p T^{5} - 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^3$ | \( 1 - 20 T^{2} + 231 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 2 T + 11 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 6 T + 41 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 4 T - 10 T^{2} - 80 T^{3} - 221 T^{4} - 80 p T^{5} - 10 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 8 T - 8 T^{2} + 80 T^{3} + 2239 T^{4} + 80 p T^{5} - 8 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + T - 40 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 10 T - 5 T^{2} - 190 T^{3} + 4876 T^{4} - 190 p T^{5} - 5 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 4 T - 76 T^{2} + 8 T^{3} + 5503 T^{4} + 8 p T^{5} - 76 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 12 T + 136 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 2 T + 35 T^{2} + 298 T^{3} - 2756 T^{4} + 298 p T^{5} + 35 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 4 T - 104 T^{2} - 8 T^{3} + 9703 T^{4} - 8 p T^{5} - 104 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 14 T + 19 T^{2} - 602 T^{3} + 13708 T^{4} - 602 p T^{5} + 19 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 136 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 10 T + 75 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 - 104 T^{2} + 4575 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 4 T - 67 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 16 T + 218 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.76660385511213378166333206803, −6.57507500041322332597402786402, −6.57177226286801055439911592853, −6.02919099939187648749648506160, −5.67256165786379348473394700027, −5.62015064915494750455352286617, −5.60267567417512015858758054547, −5.29845483134764434110959726166, −5.07488743235353835224645224587, −5.01545305954151297483665052821, −4.50725367015567432303227944304, −4.20470070690715993416516927814, −4.12624109501364922686091300810, −4.03421978485055092685184925300, −3.94921196887986187882549078793, −3.40519117633578954543157586756, −3.34359548975723224107693649038, −2.83106908464112750659762663849, −2.54906335586399980911431521724, −2.13883850801006158997067615313, −1.87622146881427255827401923145, −1.82513405865211832978742413065, −1.76971305521092401330458702364, −0.73755263789829648905802484506, −0.12111231279244567161002258194,
0.12111231279244567161002258194, 0.73755263789829648905802484506, 1.76971305521092401330458702364, 1.82513405865211832978742413065, 1.87622146881427255827401923145, 2.13883850801006158997067615313, 2.54906335586399980911431521724, 2.83106908464112750659762663849, 3.34359548975723224107693649038, 3.40519117633578954543157586756, 3.94921196887986187882549078793, 4.03421978485055092685184925300, 4.12624109501364922686091300810, 4.20470070690715993416516927814, 4.50725367015567432303227944304, 5.01545305954151297483665052821, 5.07488743235353835224645224587, 5.29845483134764434110959726166, 5.60267567417512015858758054547, 5.62015064915494750455352286617, 5.67256165786379348473394700027, 6.02919099939187648749648506160, 6.57177226286801055439911592853, 6.57507500041322332597402786402, 6.76660385511213378166333206803
