| L(s) = 1 | + 2.04e3·2-s + 3.49e5·3-s + 4.19e6·4-s − 4.88e7·5-s + 7.14e8·6-s + 1.80e8·7-s + 8.58e9·8-s + 9.04e10·9-s − 1.00e11·10-s + 1.46e12·12-s + 3.69e11·14-s − 1.70e13·15-s + 1.75e13·16-s + 1.85e14·18-s − 2.04e14·20-s + 6.30e13·21-s + 1.89e15·23-s + 2.99e15·24-s + 2.38e15·25-s + 2.06e16·27-s + 7.57e14·28-s − 2.21e16·29-s − 3.49e16·30-s + 3.60e16·32-s − 8.81e15·35-s + 3.79e17·36-s − 4.19e17·40-s + ⋯ |
| L(s) = 1 | + 2-s + 1.97·3-s + 4-s − 5-s + 1.97·6-s + 0.0913·7-s + 8-s + 2.88·9-s − 10-s + 1.97·12-s + 0.0913·14-s − 1.97·15-s + 16-s + 2.88·18-s − 20-s + 0.179·21-s + 1.98·23-s + 1.97·24-s + 25-s + 3.70·27-s + 0.0913·28-s − 1.81·29-s − 1.97·30-s + 32-s − 0.0913·35-s + 2.88·36-s − 40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(23-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+11) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{23}{2})\) |
\(\approx\) |
\(8.126626341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.126626341\) |
| \(L(12)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{11} T \) |
| 5 | \( 1 + p^{11} T \) |
| good | 3 | \( 1 - 349004 T + p^{22} T^{2} \) |
| 7 | \( 1 - 180570196 T + p^{22} T^{2} \) |
| 11 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 13 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 17 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 19 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 23 | \( 1 - 1893250316982244 T + p^{22} T^{2} \) |
| 29 | \( 1 + 22150816404486022 T + p^{22} T^{2} \) |
| 31 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 37 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 41 | \( 1 - 6435606414098462 T + p^{22} T^{2} \) |
| 43 | \( 1 - 1117494535796148124 T + p^{22} T^{2} \) |
| 47 | \( 1 + 2231392497520513004 T + p^{22} T^{2} \) |
| 53 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 59 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 61 | \( 1 + 64423241932531376458 T + p^{22} T^{2} \) |
| 67 | \( 1 + \)\(21\!\cdots\!84\)\( T + p^{22} T^{2} \) |
| 71 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 73 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 79 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 83 | \( 1 - \)\(22\!\cdots\!24\)\( T + p^{22} T^{2} \) |
| 89 | \( 1 + \)\(37\!\cdots\!42\)\( T + p^{22} T^{2} \) |
| 97 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
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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
−13.44792285196138697044224511065, −12.58918960598669443557043774719, −10.93182677975026538335407602604, −9.171621425235265019206838478517, −7.86191734607557586596548467897, −7.07553887185090618178229002182, −4.65263447511501508080186909094, −3.59734597352903329298911772891, −2.80830003263743537990735040214, −1.43697881899650127389287036387,
1.43697881899650127389287036387, 2.80830003263743537990735040214, 3.59734597352903329298911772891, 4.65263447511501508080186909094, 7.07553887185090618178229002182, 7.86191734607557586596548467897, 9.171621425235265019206838478517, 10.93182677975026538335407602604, 12.58918960598669443557043774719, 13.44792285196138697044224511065
