L(s) = 1 | − 2.69·2-s + 2.05·3-s + 5.23·4-s − 5-s − 5.51·6-s − 1.38·7-s − 8.70·8-s + 1.20·9-s + 2.69·10-s + 1.93·11-s + 10.7·12-s + 4.32·13-s + 3.72·14-s − 2.05·15-s + 12.9·16-s − 4.04·17-s − 3.24·18-s − 4.73·19-s − 5.23·20-s − 2.84·21-s − 5.21·22-s − 0.808·23-s − 17.8·24-s + 25-s − 11.6·26-s − 3.67·27-s − 7.25·28-s + ⋯ |
L(s) = 1 | − 1.90·2-s + 1.18·3-s + 2.61·4-s − 0.447·5-s − 2.25·6-s − 0.523·7-s − 3.07·8-s + 0.402·9-s + 0.850·10-s + 0.584·11-s + 3.10·12-s + 1.19·13-s + 0.996·14-s − 0.529·15-s + 3.23·16-s − 0.981·17-s − 0.765·18-s − 1.08·19-s − 1.17·20-s − 0.620·21-s − 1.11·22-s − 0.168·23-s − 3.64·24-s + 0.200·25-s − 2.27·26-s − 0.707·27-s − 1.37·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 1601 | \( 1 + T \) |
good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 3 | \( 1 - 2.05T + 3T^{2} \) |
| 7 | \( 1 + 1.38T + 7T^{2} \) |
| 11 | \( 1 - 1.93T + 11T^{2} \) |
| 13 | \( 1 - 4.32T + 13T^{2} \) |
| 17 | \( 1 + 4.04T + 17T^{2} \) |
| 19 | \( 1 + 4.73T + 19T^{2} \) |
| 23 | \( 1 + 0.808T + 23T^{2} \) |
| 29 | \( 1 + 2.23T + 29T^{2} \) |
| 31 | \( 1 - 0.820T + 31T^{2} \) |
| 37 | \( 1 - 3.61T + 37T^{2} \) |
| 41 | \( 1 - 5.58T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 - 6.91T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 7.26T + 59T^{2} \) |
| 61 | \( 1 + 1.38T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 + 2.52T + 71T^{2} \) |
| 73 | \( 1 - 4.31T + 73T^{2} \) |
| 79 | \( 1 + 9.69T + 79T^{2} \) |
| 83 | \( 1 + 7.24T + 83T^{2} \) |
| 89 | \( 1 + 6.26T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−7.73245617036405406917701214564, −7.20311770928811425636335333738, −6.27562768340863617409457611841, −6.08513734860752221708509197231, −4.26539092519947630280659679893, −3.59710912843319003775830708528, −2.74410746961188617391062500590, −2.12335398371334625907431407802, −1.17289537772218905251392624272, 0,
1.17289537772218905251392624272, 2.12335398371334625907431407802, 2.74410746961188617391062500590, 3.59710912843319003775830708528, 4.26539092519947630280659679893, 6.08513734860752221708509197231, 6.27562768340863617409457611841, 7.20311770928811425636335333738, 7.73245617036405406917701214564