| L(s) = 1 | − 3.44e3·3-s + 3.13e5·5-s − 2.32e6·7-s − 2.48e6·9-s − 5.52e7·11-s − 1.10e8·13-s − 1.07e9·15-s − 2.60e9·17-s + 1.95e9·19-s + 8.00e9·21-s − 2.54e10·23-s + 6.76e10·25-s + 5.79e10·27-s − 2.27e9·29-s − 1.90e11·31-s + 1.90e11·33-s − 7.28e11·35-s − 2.88e11·37-s + 3.79e11·39-s + 7.56e11·41-s − 3.54e11·43-s − 7.79e11·45-s + 6.03e12·47-s + 6.56e11·49-s + 8.95e12·51-s − 1.21e13·53-s − 1.73e13·55-s + ⋯ |
| L(s) = 1 | − 0.909·3-s + 1.79·5-s − 1.06·7-s − 0.173·9-s − 0.854·11-s − 0.487·13-s − 1.63·15-s − 1.53·17-s + 0.501·19-s + 0.969·21-s − 1.55·23-s + 2.21·25-s + 1.06·27-s − 0.0245·29-s − 1.24·31-s + 0.777·33-s − 1.91·35-s − 0.499·37-s + 0.443·39-s + 0.606·41-s − 0.198·43-s − 0.310·45-s + 1.73·47-s + 0.138·49-s + 1.39·51-s − 1.42·53-s − 1.53·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 1148 p T + p^{15} T^{2} \) |
| 5 | \( 1 - 313358 T + p^{15} T^{2} \) |
| 7 | \( 1 + 332088 p T + p^{15} T^{2} \) |
| 11 | \( 1 + 456604 p^{2} T + p^{15} T^{2} \) |
| 13 | \( 1 + 8481506 p T + p^{15} T^{2} \) |
| 17 | \( 1 + 2601428750 T + p^{15} T^{2} \) |
| 19 | \( 1 - 1952124284 T + p^{15} T^{2} \) |
| 23 | \( 1 + 25430340376 T + p^{15} T^{2} \) |
| 29 | \( 1 + 2277224202 T + p^{15} T^{2} \) |
| 31 | \( 1 + 190667257120 T + p^{15} T^{2} \) |
| 37 | \( 1 + 288229450002 T + p^{15} T^{2} \) |
| 41 | \( 1 - 756412456602 T + p^{15} T^{2} \) |
| 43 | \( 1 + 354186592988 T + p^{15} T^{2} \) |
| 47 | \( 1 - 6035922573648 T + p^{15} T^{2} \) |
| 53 | \( 1 + 12198920684962 T + p^{15} T^{2} \) |
| 59 | \( 1 + 4090911936748 T + p^{15} T^{2} \) |
| 61 | \( 1 - 17565907389910 T + p^{15} T^{2} \) |
| 67 | \( 1 + 3931246965172 T + p^{15} T^{2} \) |
| 71 | \( 1 - 58825436072248 T + p^{15} T^{2} \) |
| 73 | \( 1 - 107571519617914 T + p^{15} T^{2} \) |
| 79 | \( 1 - 61543860115504 T + p^{15} T^{2} \) |
| 83 | \( 1 - 13432070277436 T + p^{15} T^{2} \) |
| 89 | \( 1 - 269696339030634 T + p^{15} T^{2} \) |
| 97 | \( 1 + 793796744596318 T + p^{15} T^{2} \) |
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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
−17.35916726719619207729772310319, −16.08891511113012821856915353691, −13.86482837944641399267169775448, −12.68757426044401664695802347514, −10.63774881811528657140241443821, −9.422760105274645794954149006873, −6.45750326940849579259429532906, −5.41777306936242744170831132313, −2.34819292753910325777657924992, 0,
2.34819292753910325777657924992, 5.41777306936242744170831132313, 6.45750326940849579259429532906, 9.422760105274645794954149006873, 10.63774881811528657140241443821, 12.68757426044401664695802347514, 13.86482837944641399267169775448, 16.08891511113012821856915353691, 17.35916726719619207729772310319
