| L(s) = 1 | − 2.41·2-s + 3.82·4-s + 5-s − 1.41·7-s − 4.41·8-s − 2.41·10-s + 2.24·11-s − 3.41·13-s + 3.41·14-s + 2.99·16-s − 1.17·17-s − 19-s + 3.82·20-s − 5.41·22-s − 7.65·23-s + 25-s + 8.24·26-s − 5.41·28-s − 1.41·29-s − 3.17·31-s + 1.58·32-s + 2.82·34-s − 1.41·35-s − 3.41·37-s + 2.41·38-s − 4.41·40-s + 0.242·41-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 1.91·4-s + 0.447·5-s − 0.534·7-s − 1.56·8-s − 0.763·10-s + 0.676·11-s − 0.946·13-s + 0.912·14-s + 0.749·16-s − 0.284·17-s − 0.229·19-s + 0.856·20-s − 1.15·22-s − 1.59·23-s + 0.200·25-s + 1.61·26-s − 1.02·28-s − 0.262·29-s − 0.569·31-s + 0.280·32-s + 0.485·34-s − 0.239·35-s − 0.561·37-s + 0.391·38-s − 0.697·40-s + 0.0378·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 - 2.24T + 11T^{2} \) |
| 13 | \( 1 + 3.41T + 13T^{2} \) |
| 17 | \( 1 + 1.17T + 17T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 + 3.17T + 31T^{2} \) |
| 37 | \( 1 + 3.41T + 37T^{2} \) |
| 41 | \( 1 - 0.242T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 - 7.65T + 47T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 7.31T + 61T^{2} \) |
| 67 | \( 1 + 9.65T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 + 7.65T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 5.89T + 89T^{2} \) |
| 97 | \( 1 + 9.75T + 97T^{2} \) |
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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
−9.514531458906954383141396676411, −9.227281111221382192476111583475, −8.187536557251497965678579957603, −7.37133136499828317476175292112, −6.59797370219109302594742518380, −5.76376858388568670659294503497, −4.19892764820537397559298863593, −2.66266741777911759740479304679, −1.65084733236590813945970926393, 0,
1.65084733236590813945970926393, 2.66266741777911759740479304679, 4.19892764820537397559298863593, 5.76376858388568670659294503497, 6.59797370219109302594742518380, 7.37133136499828317476175292112, 8.187536557251497965678579957603, 9.227281111221382192476111583475, 9.514531458906954383141396676411
