| L(s) = 1 | − 2.24·2-s + 3-s + 3.03·4-s + 5-s − 2.24·6-s + 3.68·7-s − 2.32·8-s + 9-s − 2.24·10-s − 5.85·11-s + 3.03·12-s − 13-s − 8.27·14-s + 15-s − 0.849·16-s + 4.18·17-s − 2.24·18-s − 6.31·19-s + 3.03·20-s + 3.68·21-s + 13.1·22-s − 6.24·23-s − 2.32·24-s + 25-s + 2.24·26-s + 27-s + 11.1·28-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 0.577·3-s + 1.51·4-s + 0.447·5-s − 0.916·6-s + 1.39·7-s − 0.823·8-s + 0.333·9-s − 0.709·10-s − 1.76·11-s + 0.876·12-s − 0.277·13-s − 2.21·14-s + 0.258·15-s − 0.212·16-s + 1.01·17-s − 0.529·18-s − 1.44·19-s + 0.679·20-s + 0.804·21-s + 2.80·22-s − 1.30·23-s − 0.475·24-s + 0.200·25-s + 0.440·26-s + 0.192·27-s + 2.11·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 2.24T + 2T^{2} \) |
| 7 | \( 1 - 3.68T + 7T^{2} \) |
| 11 | \( 1 + 5.85T + 11T^{2} \) |
| 17 | \( 1 - 4.18T + 17T^{2} \) |
| 19 | \( 1 + 6.31T + 19T^{2} \) |
| 23 | \( 1 + 6.24T + 23T^{2} \) |
| 29 | \( 1 + 2.86T + 29T^{2} \) |
| 37 | \( 1 - 3.60T + 37T^{2} \) |
| 41 | \( 1 + 4.16T + 41T^{2} \) |
| 43 | \( 1 - 4.81T + 43T^{2} \) |
| 47 | \( 1 - 9.19T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 3.44T + 59T^{2} \) |
| 61 | \( 1 + 7.47T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 + 9.03T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 + 3.17T + 97T^{2} \) |
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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
−7.956995970041250618339559420635, −7.60724549746130628779996088315, −6.59926318786475444058177068268, −5.61782090407848765707107883698, −4.93512678663117921663179368455, −4.01692592685457982513145744947, −2.57013546153130778547545625346, −2.17295093277087711573649836314, −1.35325084385881051905563468079, 0,
1.35325084385881051905563468079, 2.17295093277087711573649836314, 2.57013546153130778547545625346, 4.01692592685457982513145744947, 4.93512678663117921663179368455, 5.61782090407848765707107883698, 6.59926318786475444058177068268, 7.60724549746130628779996088315, 7.956995970041250618339559420635
