| L(s) = 1 | − 2.04e3·2-s − 1.99e5·3-s + 4.19e6·4-s + 4.07e8·6-s − 8.58e9·8-s + 8.24e9·9-s − 5.50e11·11-s − 8.34e11·12-s + 1.75e13·16-s − 4.13e13·17-s − 1.68e13·18-s − 8.64e13·19-s + 1.12e15·22-s + 1.70e15·24-s + 2.38e15·25-s + 4.60e15·27-s − 3.60e16·32-s + 1.09e17·33-s + 8.46e16·34-s + 3.45e16·36-s + 1.77e17·38-s + 2.91e17·41-s − 1.81e18·43-s − 2.30e18·44-s − 3.50e18·48-s + 3.90e18·49-s − 4.88e18·50-s + ⋯ |
| L(s) = 1 | − 2-s − 1.12·3-s + 4-s + 1.12·6-s − 8-s + 0.262·9-s − 1.92·11-s − 1.12·12-s + 16-s − 1.20·17-s − 0.262·18-s − 0.742·19-s + 1.92·22-s + 1.12·24-s + 25-s + 0.828·27-s − 32-s + 2.16·33-s + 1.20·34-s + 0.262·36-s + 0.742·38-s + 0.529·41-s − 1.94·43-s − 1.92·44-s − 1.12·48-s + 49-s − 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(23-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+11) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{23}{2})\) |
\(\approx\) |
\(0.3522203987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3522203987\) |
| \(L(12)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{11} T \) |
| good | 3 | \( 1 + 199058 T + p^{22} T^{2} \) |
| 5 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 7 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 11 | \( 1 + 550486028386 T + p^{22} T^{2} \) |
| 13 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 17 | \( 1 + 41341538741182 T + p^{22} T^{2} \) |
| 19 | \( 1 + 86441697072754 T + p^{22} T^{2} \) |
| 23 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 29 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 31 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 37 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 41 | \( 1 - 291475944231948914 T + p^{22} T^{2} \) |
| 43 | \( 1 + 1810527321484332514 T + p^{22} T^{2} \) |
| 47 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 53 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 59 | \( 1 - 49867838021809381118 T + p^{22} T^{2} \) |
| 61 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 67 | \( 1 - \)\(20\!\cdots\!78\)\( T + p^{22} T^{2} \) |
| 71 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 73 | \( 1 - \)\(53\!\cdots\!62\)\( T + p^{22} T^{2} \) |
| 79 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 83 | \( 1 + \)\(24\!\cdots\!58\)\( T + p^{22} T^{2} \) |
| 89 | \( 1 - \)\(50\!\cdots\!26\)\( T + p^{22} T^{2} \) |
| 97 | \( 1 + \)\(94\!\cdots\!06\)\( T + p^{22} T^{2} \) |
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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
−16.51934727995361762674984703097, −15.39030312816866866960401977996, −12.80352981437104968172643691416, −11.21771251343653378618836829975, −10.36392655492652221089458785323, −8.401272461348177170608005200558, −6.72480349901271755695849527783, −5.25173405044137644871211994545, −2.44572912472585422590700815399, −0.45062506713439186006058141062,
0.45062506713439186006058141062, 2.44572912472585422590700815399, 5.25173405044137644871211994545, 6.72480349901271755695849527783, 8.401272461348177170608005200558, 10.36392655492652221089458785323, 11.21771251343653378618836829975, 12.80352981437104968172643691416, 15.39030312816866866960401977996, 16.51934727995361762674984703097
