| L(s) = 1 | + 6·3-s − 4·5-s − 16·7-s − 2·11-s − 28·13-s − 24·15-s − 66·17-s − 66·19-s − 96·21-s − 176·23-s − 184·25-s − 50·27-s + 87·29-s + 190·31-s − 12·33-s + 64·35-s − 442·37-s − 168·39-s + 1.16e3·41-s + 30·43-s + 738·47-s + 39·49-s − 396·51-s − 312·53-s + 8·55-s − 396·57-s + 44·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.357·5-s − 0.863·7-s − 0.0548·11-s − 0.597·13-s − 0.413·15-s − 0.941·17-s − 0.796·19-s − 0.997·21-s − 1.59·23-s − 1.47·25-s − 0.356·27-s + 0.557·29-s + 1.10·31-s − 0.0633·33-s + 0.309·35-s − 1.96·37-s − 0.689·39-s + 4.42·41-s + 0.106·43-s + 2.29·47-s + 0.113·49-s − 1.08·51-s − 0.808·53-s + 0.0196·55-s − 0.920·57-s + 0.0970·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.484165847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.484165847\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 2 p T + 4 p^{2} T^{2} - 166 T^{3} + 4 p^{5} T^{4} - 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 4 T + 8 p^{2} T^{2} + 614 T^{3} + 8 p^{5} T^{4} + 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 16 T + 31 p T^{2} - 2480 T^{3} + 31 p^{4} T^{4} + 16 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 3692 T^{2} + 5218 T^{3} + 3692 p^{3} T^{4} + 2 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 28 T + 2592 T^{2} - 14486 T^{3} + 2592 p^{3} T^{4} + 28 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 66 T + 14555 T^{2} + 646820 T^{3} + 14555 p^{3} T^{4} + 66 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 66 T + 10961 T^{2} + 511756 T^{3} + 10961 p^{3} T^{4} + 66 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 176 T + 44693 T^{2} + 4389472 T^{3} + 44693 p^{3} T^{4} + 176 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 190 T + 34940 T^{2} - 1815354 T^{3} + 34940 p^{3} T^{4} - 190 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 442 T + 159351 T^{2} + 34453924 T^{3} + 159351 p^{3} T^{4} + 442 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 1162 T + 645399 T^{2} - 213908268 T^{3} + 645399 p^{3} T^{4} - 1162 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 30 T + 157244 T^{2} - 12805582 T^{3} + 157244 p^{3} T^{4} - 30 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 738 T + 470756 T^{2} - 163890206 T^{3} + 470756 p^{3} T^{4} - 738 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 312 T + 349160 T^{2} + 97820330 T^{3} + 349160 p^{3} T^{4} + 312 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 44 T + 592857 T^{2} - 18923016 T^{3} + 592857 p^{3} T^{4} - 44 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 54 T + 121287 T^{2} - 62852868 T^{3} + 121287 p^{3} T^{4} + 54 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 116 T + 720193 T^{2} + 88800824 T^{3} + 720193 p^{3} T^{4} + 116 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 1200 T + 1511489 T^{2} - 903051776 T^{3} + 1511489 p^{3} T^{4} - 1200 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 1118 T + 1314795 T^{2} + 850221500 T^{3} + 1314795 p^{3} T^{4} + 1118 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 2262 T + 36404 p T^{2} - 2430107338 T^{3} + 36404 p^{4} T^{4} - 2262 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 1804 T + 1839409 T^{2} + 1410713352 T^{3} + 1839409 p^{3} T^{4} + 1804 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 1578 T + 1639271 T^{2} - 1200535660 T^{3} + 1639271 p^{3} T^{4} - 1578 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 1450 T + 1528271 T^{2} - 826334604 T^{3} + 1528271 p^{3} T^{4} - 1450 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.072892610458205673381765737409, −7.53936517309890250646284637391, −7.42785715362284246992230849667, −7.23175514105499649766468197523, −6.66603546233126812819039032058, −6.56474840887606128564810013402, −6.25763798650350985701342882838, −5.95618962528913321289235789564, −5.73130481340176389450071768870, −5.56587091277968024880312134768, −5.04361586329087956553895756551, −4.54935948356122501099980012255, −4.33372836398159431185232956205, −4.28556878322058048937870253724, −3.76833999296329831475636297671, −3.71126617532956123693867491201, −3.07348439256872127241787591997, −2.99622771517840769566612201252, −2.56571794422073958366809009858, −2.19201475807545291725167297247, −2.09153643984546117523166341893, −1.81213968486903860629960583990, −0.834271316247086494623075309793, −0.68112918899592510934806196176, −0.25069234914285467695743834697,
0.25069234914285467695743834697, 0.68112918899592510934806196176, 0.834271316247086494623075309793, 1.81213968486903860629960583990, 2.09153643984546117523166341893, 2.19201475807545291725167297247, 2.56571794422073958366809009858, 2.99622771517840769566612201252, 3.07348439256872127241787591997, 3.71126617532956123693867491201, 3.76833999296329831475636297671, 4.28556878322058048937870253724, 4.33372836398159431185232956205, 4.54935948356122501099980012255, 5.04361586329087956553895756551, 5.56587091277968024880312134768, 5.73130481340176389450071768870, 5.95618962528913321289235789564, 6.25763798650350985701342882838, 6.56474840887606128564810013402, 6.66603546233126812819039032058, 7.23175514105499649766468197523, 7.42785715362284246992230849667, 7.53936517309890250646284637391, 8.072892610458205673381765737409
