| L(s) = 1 | + 1.12e7·5-s + 2.81e8·7-s + 3.61e10·11-s − 4.49e11·13-s − 2.12e12·17-s − 4.60e12·19-s − 9.50e13·23-s − 3.49e14·25-s + 2.24e15·29-s − 3.15e15·31-s + 3.17e15·35-s − 1.81e16·37-s + 1.69e17·41-s − 1.58e17·43-s + 1.34e17·47-s − 4.79e17·49-s + 1.56e16·53-s + 4.07e17·55-s − 2.97e18·59-s + 3.60e18·61-s − 5.06e18·65-s + 2.10e19·67-s − 2.19e19·71-s − 1.70e19·73-s + 1.01e19·77-s − 1.15e20·79-s + 9.66e19·83-s + ⋯ |
| L(s) = 1 | + 0.516·5-s + 0.377·7-s + 0.420·11-s − 0.903·13-s − 0.255·17-s − 0.172·19-s − 0.478·23-s − 0.733·25-s + 0.991·29-s − 0.691·31-s + 0.194·35-s − 0.621·37-s + 1.97·41-s − 1.12·43-s + 0.373·47-s − 0.857·49-s + 0.0122·53-s + 0.216·55-s − 0.758·59-s + 0.646·61-s − 0.466·65-s + 1.41·67-s − 0.801·71-s − 0.464·73-s + 0.158·77-s − 1.36·79-s + 0.683·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2253618 p T + p^{21} T^{2} \) |
| 7 | \( 1 - 40273448 p T + p^{21} T^{2} \) |
| 11 | \( 1 - 36172082484 T + p^{21} T^{2} \) |
| 13 | \( 1 + 34546044490 p T + p^{21} T^{2} \) |
| 17 | \( 1 + 124815222738 p T + p^{21} T^{2} \) |
| 19 | \( 1 + 242600328100 p T + p^{21} T^{2} \) |
| 23 | \( 1 + 95095276921656 T + p^{21} T^{2} \) |
| 29 | \( 1 - 77439392529354 p T + p^{21} T^{2} \) |
| 31 | \( 1 + 3155693201792656 T + p^{21} T^{2} \) |
| 37 | \( 1 + 18178503074861482 T + p^{21} T^{2} \) |
| 41 | \( 1 - 169649739387485910 T + p^{21} T^{2} \) |
| 43 | \( 1 + 158968551608988244 T + p^{21} T^{2} \) |
| 47 | \( 1 - 134697468442682736 T + p^{21} T^{2} \) |
| 53 | \( 1 - 15637375269722538 T + p^{21} T^{2} \) |
| 59 | \( 1 + 2977241337691499484 T + p^{21} T^{2} \) |
| 61 | \( 1 - 3603855625679330702 T + p^{21} T^{2} \) |
| 67 | \( 1 - 21066199531967164004 T + p^{21} T^{2} \) |
| 71 | \( 1 + 21980089544074358760 T + p^{21} T^{2} \) |
| 73 | \( 1 + 17054415965500339222 T + p^{21} T^{2} \) |
| 79 | \( 1 + \)\(11\!\cdots\!52\)\( T + p^{21} T^{2} \) |
| 83 | \( 1 - 96628520442403345644 T + p^{21} T^{2} \) |
| 89 | \( 1 + 60427571095732966650 T + p^{21} T^{2} \) |
| 97 | \( 1 + \)\(40\!\cdots\!98\)\( T + p^{21} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.53183177958976774579025115426, −10.23684728934094697476540658045, −9.210082324324547376402380979535, −7.88844311851683927771994312137, −6.60067166877627645642040037460, −5.35574721960332546264241862071, −4.13582179178237159348181735536, −2.55606581989517353341033685707, −1.48407631671678202281933849285, 0,
1.48407631671678202281933849285, 2.55606581989517353341033685707, 4.13582179178237159348181735536, 5.35574721960332546264241862071, 6.60067166877627645642040037460, 7.88844311851683927771994312137, 9.210082324324547376402380979535, 10.23684728934094697476540658045, 11.53183177958976774579025115426
