| L(s) = 1 | − 7.25·2-s − 9·3-s + 20.6·4-s + 11.4·5-s + 65.2·6-s + 191.·7-s + 82.6·8-s + 81·9-s − 83.0·10-s − 523.·11-s − 185.·12-s − 192.·13-s − 1.38e3·14-s − 103.·15-s − 1.25e3·16-s − 1.25e3·17-s − 587.·18-s + 966.·19-s + 235.·20-s − 1.72e3·21-s + 3.80e3·22-s + 2.69e3·23-s − 743.·24-s − 2.99e3·25-s + 1.39e3·26-s − 729·27-s + 3.94e3·28-s + ⋯ |
| L(s) = 1 | − 1.28·2-s − 0.577·3-s + 0.644·4-s + 0.204·5-s + 0.740·6-s + 1.47·7-s + 0.456·8-s + 0.333·9-s − 0.262·10-s − 1.30·11-s − 0.371·12-s − 0.315·13-s − 1.89·14-s − 0.118·15-s − 1.22·16-s − 1.05·17-s − 0.427·18-s + 0.614·19-s + 0.131·20-s − 0.851·21-s + 1.67·22-s + 1.06·23-s − 0.263·24-s − 0.958·25-s + 0.404·26-s − 0.192·27-s + 0.950·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 9T \) |
| 103 | \( 1 + 1.06e4T \) |
| good | 2 | \( 1 + 7.25T + 32T^{2} \) |
| 5 | \( 1 - 11.4T + 3.12e3T^{2} \) |
| 7 | \( 1 - 191.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 523.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 192.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.25e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 966.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.69e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.82e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.04e4T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.52e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.79e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.68e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.37e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.60e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.35e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.07e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.07e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.44e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.33e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.56e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.01e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.63e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.85e3T + 8.58e9T^{2} \) |
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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
−10.35026202000985261211293613268, −9.534166571783751403848628516314, −8.318991588678777048647207320419, −7.84984344322726352169363497195, −6.78966361106883785294579844434, −5.23456096568819841794128957110, −4.61374042948003372429873216718, −2.35383209591487648302656588780, −1.24362654331488841502473075001, 0,
1.24362654331488841502473075001, 2.35383209591487648302656588780, 4.61374042948003372429873216718, 5.23456096568819841794128957110, 6.78966361106883785294579844434, 7.84984344322726352169363497195, 8.318991588678777048647207320419, 9.534166571783751403848628516314, 10.35026202000985261211293613268
