| L(s) = 1 | + 3.31·2-s + 7.10·3-s + 2.98·4-s + 23.5·6-s − 9.84·7-s − 16.6·8-s + 23.4·9-s + 0.597·11-s + 21.2·12-s − 34.9·13-s − 32.6·14-s − 78.9·16-s − 91.3·17-s + 77.7·18-s − 24.0·19-s − 69.9·21-s + 1.98·22-s − 127.·23-s − 118.·24-s − 115.·26-s − 25.0·27-s − 29.3·28-s + 285.·29-s − 192.·31-s − 128.·32-s + 4.24·33-s − 302.·34-s + ⋯ |
| L(s) = 1 | + 1.17·2-s + 1.36·3-s + 0.373·4-s + 1.60·6-s − 0.531·7-s − 0.734·8-s + 0.869·9-s + 0.0163·11-s + 0.510·12-s − 0.744·13-s − 0.622·14-s − 1.23·16-s − 1.30·17-s + 1.01·18-s − 0.290·19-s − 0.726·21-s + 0.0192·22-s − 1.15·23-s − 1.00·24-s − 0.872·26-s − 0.178·27-s − 0.198·28-s + 1.82·29-s − 1.11·31-s − 0.711·32-s + 0.0224·33-s − 1.52·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 - 43T \) |
| good | 2 | \( 1 - 3.31T + 8T^{2} \) |
| 3 | \( 1 - 7.10T + 27T^{2} \) |
| 7 | \( 1 + 9.84T + 343T^{2} \) |
| 11 | \( 1 - 0.597T + 1.33e3T^{2} \) |
| 13 | \( 1 + 34.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 91.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 24.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 127.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 285.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 192.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 363.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 191.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 458.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 583.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 416.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 30.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 16.1T + 3.00e5T^{2} \) |
| 71 | \( 1 + 219.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.03e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 445.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 298.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 846.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 259.T + 9.12e5T^{2} \) |
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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
−8.942377991411315974314784008602, −8.476266029254385899008433434794, −7.29984670426113132909015124848, −6.52004707263421478624037122233, −5.50384815360254681999701078080, −4.39727027694383044855489498001, −3.79388220109862363546538067817, −2.79385885016001116958412245606, −2.17451658745586945030337030638, 0,
2.17451658745586945030337030638, 2.79385885016001116958412245606, 3.79388220109862363546538067817, 4.39727027694383044855489498001, 5.50384815360254681999701078080, 6.52004707263421478624037122233, 7.29984670426113132909015124848, 8.476266029254385899008433434794, 8.942377991411315974314784008602
