L(s) = 1 | − 0.217·2-s − 2.35·3-s − 1.95·4-s + 5-s + 0.514·6-s + 7-s + 0.861·8-s + 2.56·9-s − 0.217·10-s − 4.74·11-s + 4.60·12-s − 3.63·13-s − 0.217·14-s − 2.35·15-s + 3.71·16-s − 4.50·17-s − 0.559·18-s − 0.110·19-s − 1.95·20-s − 2.35·21-s + 1.03·22-s + 6.89·23-s − 2.03·24-s + 25-s + 0.792·26-s + 1.01·27-s − 1.95·28-s + ⋯ |
L(s) = 1 | − 0.154·2-s − 1.36·3-s − 0.976·4-s + 0.447·5-s + 0.209·6-s + 0.377·7-s + 0.304·8-s + 0.856·9-s − 0.0688·10-s − 1.42·11-s + 1.33·12-s − 1.00·13-s − 0.0582·14-s − 0.609·15-s + 0.929·16-s − 1.09·17-s − 0.131·18-s − 0.0253·19-s − 0.436·20-s − 0.514·21-s + 0.220·22-s + 1.43·23-s − 0.414·24-s + 0.200·25-s + 0.155·26-s + 0.195·27-s − 0.368·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
good | 2 | \( 1 + 0.217T + 2T^{2} \) |
| 3 | \( 1 + 2.35T + 3T^{2} \) |
| 11 | \( 1 + 4.74T + 11T^{2} \) |
| 13 | \( 1 + 3.63T + 13T^{2} \) |
| 17 | \( 1 + 4.50T + 17T^{2} \) |
| 19 | \( 1 + 0.110T + 19T^{2} \) |
| 23 | \( 1 - 6.89T + 23T^{2} \) |
| 29 | \( 1 + 2.25T + 29T^{2} \) |
| 31 | \( 1 - 1.63T + 31T^{2} \) |
| 37 | \( 1 + 2.55T + 37T^{2} \) |
| 41 | \( 1 - 3.68T + 41T^{2} \) |
| 43 | \( 1 + 2.34T + 43T^{2} \) |
| 47 | \( 1 + 5.07T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 - 9.43T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 6.08T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 4.93T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 - 0.0568T + 89T^{2} \) |
| 97 | \( 1 - 7.57T + 97T^{2} \) |
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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
−7.23584299224009105378021733299, −6.94875945612559702277438048000, −5.68862232776739150647477875870, −5.47686847594478380831580507430, −4.76753977882283125766808248408, −4.41046833708477838626873672555, −3.03204360238152151455390058038, −2.11963608813962723521088537869, −0.838986504498371251312124127110, 0,
0.838986504498371251312124127110, 2.11963608813962723521088537869, 3.03204360238152151455390058038, 4.41046833708477838626873672555, 4.76753977882283125766808248408, 5.47686847594478380831580507430, 5.68862232776739150647477875870, 6.94875945612559702277438048000, 7.23584299224009105378021733299