| L(s) = 1 | − 3.39e8·3-s + 6.87e10·4-s + 1.75e12·5-s − 3.49e16·9-s + 5.55e18·11-s − 2.33e19·12-s − 5.95e20·15-s + 4.72e21·16-s + 1.20e23·20-s + 6.42e24·23-s − 1.14e25·25-s + 6.27e25·27-s − 6.88e26·31-s − 1.88e27·33-s − 2.40e27·36-s − 3.36e28·37-s + 3.82e29·44-s − 6.14e28·45-s + 1.96e30·47-s − 1.60e30·48-s + 2.65e30·49-s − 1.99e31·53-s + 9.76e30·55-s + 3.02e30·59-s − 4.09e31·60-s + 3.24e32·64-s − 6.55e31·67-s + ⋯ |
| L(s) = 1 | − 0.875·3-s + 4-s + 0.460·5-s − 0.233·9-s + 11-s − 0.875·12-s − 0.403·15-s + 16-s + 0.460·20-s + 1.97·23-s − 0.788·25-s + 1.07·27-s − 0.985·31-s − 0.875·33-s − 0.233·36-s − 1.99·37-s + 44-s − 0.107·45-s + 1.56·47-s − 0.875·48-s + 49-s − 1.82·53-s + 0.460·55-s + 0.0402·59-s − 0.403·60-s + 64-s − 0.0885·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(37-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+18) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{37}{2})\) |
\(\approx\) |
\(2.388094201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.388094201\) |
| \(L(19)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - p^{18} T \) |
| good | 2 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 3 | \( 1 + 339284078 T + p^{36} T^{2} \) |
| 5 | \( 1 - 1755797021426 T + p^{36} T^{2} \) |
| 7 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 13 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 17 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 19 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 23 | \( 1 - \)\(64\!\cdots\!62\)\( T + p^{36} T^{2} \) |
| 29 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 31 | \( 1 + \)\(68\!\cdots\!18\)\( T + p^{36} T^{2} \) |
| 37 | \( 1 + \)\(33\!\cdots\!58\)\( T + p^{36} T^{2} \) |
| 41 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 43 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 47 | \( 1 - \)\(19\!\cdots\!22\)\( T + p^{36} T^{2} \) |
| 53 | \( 1 + \)\(19\!\cdots\!78\)\( T + p^{36} T^{2} \) |
| 59 | \( 1 - \)\(30\!\cdots\!42\)\( T + p^{36} T^{2} \) |
| 61 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 67 | \( 1 + \)\(65\!\cdots\!18\)\( T + p^{36} T^{2} \) |
| 71 | \( 1 - \)\(41\!\cdots\!22\)\( T + p^{36} T^{2} \) |
| 73 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 79 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 83 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 89 | \( 1 - \)\(14\!\cdots\!62\)\( T + p^{36} T^{2} \) |
| 97 | \( 1 - \)\(11\!\cdots\!22\)\( T + p^{36} T^{2} \) |
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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
−12.47277876000139659720910353511, −11.46128463202050685322805489804, −10.60212743812085921860339386503, −9.022092826351267372847408091727, −7.16315774511196555608312471030, −6.22261455438243884114256967554, −5.22347811326116481846471573719, −3.37083331353773215414363024582, −1.94733890231019896937495340838, −0.811796754892330039654942018097,
0.811796754892330039654942018097, 1.94733890231019896937495340838, 3.37083331353773215414363024582, 5.22347811326116481846471573719, 6.22261455438243884114256967554, 7.16315774511196555608312471030, 9.022092826351267372847408091727, 10.60212743812085921860339386503, 11.46128463202050685322805489804, 12.47277876000139659720910353511
