| L(s) = 1 | + 1.67·3-s − 5-s − 2.58·7-s − 0.179·9-s + 4.59·11-s + 6.44·13-s − 1.67·15-s + 3.22·17-s − 7.28·19-s − 4.33·21-s − 7.91·23-s + 25-s − 5.33·27-s − 9.25·29-s − 3.08·31-s + 7.71·33-s + 2.58·35-s + 2.24·37-s + 10.8·39-s + 2.13·41-s + 7.12·43-s + 0.179·45-s − 1.12·47-s − 0.336·49-s + 5.41·51-s − 13.2·53-s − 4.59·55-s + ⋯ |
| L(s) = 1 | + 0.969·3-s − 0.447·5-s − 0.975·7-s − 0.0597·9-s + 1.38·11-s + 1.78·13-s − 0.433·15-s + 0.781·17-s − 1.67·19-s − 0.946·21-s − 1.65·23-s + 0.200·25-s − 1.02·27-s − 1.71·29-s − 0.554·31-s + 1.34·33-s + 0.436·35-s + 0.368·37-s + 1.73·39-s + 0.333·41-s + 1.08·43-s + 0.0266·45-s − 0.163·47-s − 0.0481·49-s + 0.758·51-s − 1.81·53-s − 0.619·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 1.67T + 3T^{2} \) |
| 7 | \( 1 + 2.58T + 7T^{2} \) |
| 11 | \( 1 - 4.59T + 11T^{2} \) |
| 13 | \( 1 - 6.44T + 13T^{2} \) |
| 17 | \( 1 - 3.22T + 17T^{2} \) |
| 19 | \( 1 + 7.28T + 19T^{2} \) |
| 23 | \( 1 + 7.91T + 23T^{2} \) |
| 29 | \( 1 + 9.25T + 29T^{2} \) |
| 31 | \( 1 + 3.08T + 31T^{2} \) |
| 37 | \( 1 - 2.24T + 37T^{2} \) |
| 41 | \( 1 - 2.13T + 41T^{2} \) |
| 43 | \( 1 - 7.12T + 43T^{2} \) |
| 47 | \( 1 + 1.12T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 - 3.97T + 59T^{2} \) |
| 61 | \( 1 - 8.25T + 61T^{2} \) |
| 67 | \( 1 - 0.637T + 67T^{2} \) |
| 71 | \( 1 + 3.08T + 71T^{2} \) |
| 73 | \( 1 + 5.63T + 73T^{2} \) |
| 79 | \( 1 - 8.87T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 - 5.95T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−7.75812677535637324870493712593, −6.66544668622225753485101492104, −6.20350170640871369889089822618, −5.68599224558122260186169128537, −4.08364923274974300038630596623, −3.84064930265631564879871982032, −3.37591707878431819354060233816, −2.27019925582061829208394311903, −1.41402069269485335909508544716, 0,
1.41402069269485335909508544716, 2.27019925582061829208394311903, 3.37591707878431819354060233816, 3.84064930265631564879871982032, 4.08364923274974300038630596623, 5.68599224558122260186169128537, 6.20350170640871369889089822618, 6.66544668622225753485101492104, 7.75812677535637324870493712593
