| L(s) = 1 | + 1.76·2-s + 2.03·3-s + 1.10·4-s − 5-s + 3.59·6-s − 1.48·7-s − 1.57·8-s + 1.15·9-s − 1.76·10-s + 11-s + 2.24·12-s + 2.46·13-s − 2.61·14-s − 2.03·15-s − 4.98·16-s − 7.35·17-s + 2.03·18-s − 5.22·19-s − 1.10·20-s − 3.02·21-s + 1.76·22-s − 3.91·23-s − 3.21·24-s + 25-s + 4.34·26-s − 3.76·27-s − 1.64·28-s + ⋯ |
| L(s) = 1 | + 1.24·2-s + 1.17·3-s + 0.551·4-s − 0.447·5-s + 1.46·6-s − 0.561·7-s − 0.558·8-s + 0.384·9-s − 0.557·10-s + 0.301·11-s + 0.649·12-s + 0.684·13-s − 0.699·14-s − 0.526·15-s − 1.24·16-s − 1.78·17-s + 0.479·18-s − 1.19·19-s − 0.246·20-s − 0.660·21-s + 0.375·22-s − 0.817·23-s − 0.656·24-s + 0.200·25-s + 0.852·26-s − 0.724·27-s − 0.310·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 - 1.76T + 2T^{2} \) |
| 3 | \( 1 - 2.03T + 3T^{2} \) |
| 7 | \( 1 + 1.48T + 7T^{2} \) |
| 13 | \( 1 - 2.46T + 13T^{2} \) |
| 17 | \( 1 + 7.35T + 17T^{2} \) |
| 19 | \( 1 + 5.22T + 19T^{2} \) |
| 23 | \( 1 + 3.91T + 23T^{2} \) |
| 29 | \( 1 + 4.31T + 29T^{2} \) |
| 31 | \( 1 - 8.69T + 31T^{2} \) |
| 37 | \( 1 + 8.91T + 37T^{2} \) |
| 41 | \( 1 - 4.09T + 41T^{2} \) |
| 43 | \( 1 - 7.67T + 43T^{2} \) |
| 47 | \( 1 - 2.95T + 47T^{2} \) |
| 53 | \( 1 + 13.0T + 53T^{2} \) |
| 59 | \( 1 - 0.524T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 - 8.33T + 67T^{2} \) |
| 71 | \( 1 + 9.41T + 71T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 - 5.14T + 97T^{2} \) |
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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
−8.301101428592788708700449753900, −7.20287915067935234713749210357, −6.40862118098450932507416304198, −5.97474437438256062797412013604, −4.71639029065292091286549589942, −4.07237115592389508631661435975, −3.61954862176703194464691965852, −2.73105972457581933847815813835, −2.04376417637980747875525747772, 0,
2.04376417637980747875525747772, 2.73105972457581933847815813835, 3.61954862176703194464691965852, 4.07237115592389508631661435975, 4.71639029065292091286549589942, 5.97474437438256062797412013604, 6.40862118098450932507416304198, 7.20287915067935234713749210357, 8.301101428592788708700449753900
