| L(s) = 1 | + 4.77e5·2-s + 2.23e7·3-s + 1.59e11·4-s + 1.06e13·6-s + 4.33e16·8-s − 1.49e17·9-s + 3.55e18·12-s − 1.75e19·13-s + 9.75e21·16-s − 7.14e22·18-s + 3.24e24·23-s + 9.67e23·24-s + 1.45e25·25-s − 8.36e24·26-s − 6.68e24·27-s + 2.44e26·29-s + 7.53e26·31-s + 1.68e27·32-s − 2.38e28·36-s − 3.90e26·39-s + 7.58e28·41-s + 1.54e30·46-s + 1.56e30·47-s + 2.17e29·48-s + 2.65e30·49-s + 6.95e30·50-s − 2.79e30·52-s + ⋯ |
| L(s) = 1 | + 1.82·2-s + 0.0576·3-s + 2.32·4-s + 0.104·6-s + 2.40·8-s − 0.996·9-s + 0.133·12-s − 0.155·13-s + 2.06·16-s − 1.81·18-s + 23-s + 0.138·24-s + 25-s − 0.283·26-s − 0.115·27-s + 1.16·29-s + 1.07·31-s + 1.35·32-s − 2.31·36-s − 0.00897·39-s + 0.707·41-s + 1.82·46-s + 1.25·47-s + 0.118·48-s + 49-s + 1.82·50-s − 0.361·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(37-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+18) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{37}{2})\) |
\(\approx\) |
\(8.267910745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.267910745\) |
| \(L(19)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - p^{18} T \) |
| good | 2 | \( 1 - 477713 T + p^{36} T^{2} \) |
| 3 | \( 1 - 22317778 T + p^{36} T^{2} \) |
| 5 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 7 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 11 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 13 | \( 1 + 17519171692091752942 T + p^{36} T^{2} \) |
| 17 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 19 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 29 | \( 1 - \)\(24\!\cdots\!22\)\( T + p^{36} T^{2} \) |
| 31 | \( 1 - \)\(75\!\cdots\!82\)\( T + p^{36} T^{2} \) |
| 37 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 41 | \( 1 - \)\(75\!\cdots\!42\)\( T + p^{36} T^{2} \) |
| 43 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 47 | \( 1 - \)\(15\!\cdots\!78\)\( T + p^{36} T^{2} \) |
| 53 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 59 | \( 1 - \)\(98\!\cdots\!42\)\( T + p^{36} T^{2} \) |
| 61 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 67 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 71 | \( 1 + \)\(25\!\cdots\!78\)\( T + p^{36} T^{2} \) |
| 73 | \( 1 + \)\(42\!\cdots\!62\)\( T + p^{36} T^{2} \) |
| 79 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 83 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 89 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 97 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
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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
−11.55516694735848680920158036832, −10.53858599781318142096775522674, −8.692157141940750063673194866294, −7.19945902775418142177760613240, −6.17141658585887847833051616681, −5.22828201496141633946889199084, −4.30750723943343783718745352715, −3.04870247115625236430444582489, −2.49030585693932560257135181225, −0.929182917058695537033210808664,
0.929182917058695537033210808664, 2.49030585693932560257135181225, 3.04870247115625236430444582489, 4.30750723943343783718745352715, 5.22828201496141633946889199084, 6.17141658585887847833051616681, 7.19945902775418142177760613240, 8.692157141940750063673194866294, 10.53858599781318142096775522674, 11.55516694735848680920158036832
