L(s) = 1 | + 8.86·2-s − 1.50·3-s + 46.5·4-s − 13.3·6-s + 124.·7-s + 129.·8-s − 240.·9-s + 590.·11-s − 69.9·12-s − 434.·13-s + 1.10e3·14-s − 343.·16-s − 1.92e3·17-s − 2.13e3·18-s + 654.·19-s − 187.·21-s + 5.23e3·22-s − 2.80e3·23-s − 194.·24-s − 3.85e3·26-s + 726.·27-s + 5.81e3·28-s − 1.45e3·29-s + 4.41e3·31-s − 7.18e3·32-s − 886.·33-s − 1.70e4·34-s + ⋯ |
L(s) = 1 | + 1.56·2-s − 0.0963·3-s + 1.45·4-s − 0.150·6-s + 0.962·7-s + 0.714·8-s − 0.990·9-s + 1.47·11-s − 0.140·12-s − 0.713·13-s + 1.50·14-s − 0.335·16-s − 1.61·17-s − 1.55·18-s + 0.415·19-s − 0.0926·21-s + 2.30·22-s − 1.10·23-s − 0.0688·24-s − 1.11·26-s + 0.191·27-s + 1.40·28-s − 0.321·29-s + 0.826·31-s − 1.24·32-s − 0.141·33-s − 2.53·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 43 | \( 1 + 1.84e3T \) |
good | 2 | \( 1 - 8.86T + 32T^{2} \) |
| 3 | \( 1 + 1.50T + 243T^{2} \) |
| 7 | \( 1 - 124.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 590.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 434.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.92e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 654.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.80e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.45e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.41e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.75e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.97e3T + 1.15e8T^{2} \) |
| 47 | \( 1 + 2.20e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.49e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.27e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.10e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.52e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.80e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.88e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.27e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.84e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.03e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.88e4T + 8.58e9T^{2} \) |
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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
−8.732492967710803163826897321156, −7.74316987142740745775779107621, −6.59258315352168739782150098287, −6.13905400642525448999096482792, −5.05220514834546067250698491660, −4.50762848832024827923458217893, −3.64305533971625001759911613970, −2.54345866156770410617192414318, −1.66772008976538036820832934375, 0,
1.66772008976538036820832934375, 2.54345866156770410617192414318, 3.64305533971625001759911613970, 4.50762848832024827923458217893, 5.05220514834546067250698491660, 6.13905400642525448999096482792, 6.59258315352168739782150098287, 7.74316987142740745775779107621, 8.732492967710803163826897321156