| L(s) = 1 | − 0.164·2-s − 1.43·3-s − 31.9·4-s − 37.9·5-s + 0.235·6-s − 43.8·7-s + 10.5·8-s − 240.·9-s + 6.24·10-s − 163.·11-s + 45.7·12-s + 430.·13-s + 7.21·14-s + 54.3·15-s + 1.02e3·16-s + 740.·17-s + 39.6·18-s − 916.·19-s + 1.21e3·20-s + 62.7·21-s + 26.8·22-s − 529·23-s − 15.0·24-s − 1.68e3·25-s − 70.7·26-s + 692.·27-s + 1.40e3·28-s + ⋯ |
| L(s) = 1 | − 0.0290·2-s − 0.0917·3-s − 0.999·4-s − 0.679·5-s + 0.00266·6-s − 0.338·7-s + 0.0581·8-s − 0.991·9-s + 0.0197·10-s − 0.406·11-s + 0.0917·12-s + 0.706·13-s + 0.00983·14-s + 0.0623·15-s + 0.997·16-s + 0.621·17-s + 0.0288·18-s − 0.582·19-s + 0.678·20-s + 0.0310·21-s + 0.0118·22-s − 0.208·23-s − 0.00533·24-s − 0.538·25-s − 0.0205·26-s + 0.182·27-s + 0.337·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + 529T \) |
| good | 2 | \( 1 + 0.164T + 32T^{2} \) |
| 3 | \( 1 + 1.43T + 243T^{2} \) |
| 5 | \( 1 + 37.9T + 3.12e3T^{2} \) |
| 7 | \( 1 + 43.8T + 1.68e4T^{2} \) |
| 11 | \( 1 + 163.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 430.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 740.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 916.T + 2.47e6T^{2} \) |
| 29 | \( 1 + 5.11e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.70e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.09e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.10e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.52e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.44e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.85e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.15e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.19e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.51e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.40e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.33e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.42e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.14e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.34e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.19e4T + 8.58e9T^{2} \) |
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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
−16.28130488900852758725213997978, −14.82994923139838765169869162226, −13.61784875165198485825219203222, −12.34386286650814691422551255976, −10.83507658233044476865893056574, −9.144695026783762343558798641703, −7.926176168376805307324222843550, −5.63300935067018173421652806968, −3.71137790019432399643875222060, 0,
3.71137790019432399643875222060, 5.63300935067018173421652806968, 7.926176168376805307324222843550, 9.144695026783762343558798641703, 10.83507658233044476865893056574, 12.34386286650814691422551255976, 13.61784875165198485825219203222, 14.82994923139838765169869162226, 16.28130488900852758725213997978
