| L(s) = 1 | − 2.06·2-s + 2.59·3-s + 2.25·4-s − 5-s − 5.36·6-s − 1.01·7-s − 0.533·8-s + 3.75·9-s + 2.06·10-s − 11-s + 5.87·12-s + 0.591·13-s + 2.09·14-s − 2.59·15-s − 3.41·16-s − 1.74·17-s − 7.75·18-s − 3.68·19-s − 2.25·20-s − 2.63·21-s + 2.06·22-s + 2.17·23-s − 1.38·24-s + 25-s − 1.22·26-s + 1.96·27-s − 2.29·28-s + ⋯ |
| L(s) = 1 | − 1.45·2-s + 1.50·3-s + 1.12·4-s − 0.447·5-s − 2.18·6-s − 0.383·7-s − 0.188·8-s + 1.25·9-s + 0.652·10-s − 0.301·11-s + 1.69·12-s + 0.164·13-s + 0.559·14-s − 0.671·15-s − 0.854·16-s − 0.423·17-s − 1.82·18-s − 0.845·19-s − 0.504·20-s − 0.575·21-s + 0.439·22-s + 0.453·23-s − 0.282·24-s + 0.200·25-s − 0.239·26-s + 0.378·27-s − 0.432·28-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 | \( 1 + 2.06T + 2T^{2} \) |
| 3 | \( 1 - 2.59T + 3T^{2} \) |
| 7 | \( 1 + 1.01T + 7T^{2} \) |
| 13 | \( 1 - 0.591T + 13T^{2} \) |
| 17 | \( 1 + 1.74T + 17T^{2} \) |
| 19 | \( 1 + 3.68T + 19T^{2} \) |
| 23 | \( 1 - 2.17T + 23T^{2} \) |
| 29 | \( 1 - 8.50T + 29T^{2} \) |
| 31 | \( 1 + 2.16T + 31T^{2} \) |
| 37 | \( 1 + 8.65T + 37T^{2} \) |
| 41 | \( 1 + 1.23T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 1.57T + 53T^{2} \) |
| 59 | \( 1 + 6.86T + 59T^{2} \) |
| 61 | \( 1 + 0.278T + 61T^{2} \) |
| 67 | \( 1 - 5.68T + 67T^{2} \) |
| 71 | \( 1 + 9.63T + 71T^{2} \) |
| 79 | \( 1 - 6.04T + 79T^{2} \) |
| 83 | \( 1 + 9.00T + 83T^{2} \) |
| 89 | \( 1 - 3.21T + 89T^{2} \) |
| 97 | \( 1 - 8.99T + 97T^{2} \) |
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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
−8.312590931822225275840718533281, −7.70400357027032310709785189877, −7.01431903088249166483131551007, −6.35805215221997626054681877944, −4.86999819429452506800362341402, −4.01997763852115774333237574517, −3.07951768754885991392190719400, −2.37258496707796189270908067433, −1.40712294651671847643776300636, 0,
1.40712294651671847643776300636, 2.37258496707796189270908067433, 3.07951768754885991392190719400, 4.01997763852115774333237574517, 4.86999819429452506800362341402, 6.35805215221997626054681877944, 7.01431903088249166483131551007, 7.70400357027032310709785189877, 8.312590931822225275840718533281
