| L(s) = 1 | + 3-s − 0.167·5-s − 7-s + 9-s + 1.80·11-s − 3.63·13-s − 0.167·15-s + 4.13·17-s − 5.80·19-s − 21-s + 23-s − 4.97·25-s + 27-s + 5·29-s − 0.195·31-s + 1.80·33-s + 0.167·35-s − 3.33·37-s − 3.63·39-s − 4.94·41-s − 1.30·43-s − 0.167·45-s + 3.13·47-s + 49-s + 4.13·51-s + 3.30·53-s − 0.302·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.0748·5-s − 0.377·7-s + 0.333·9-s + 0.544·11-s − 1.00·13-s − 0.0432·15-s + 1.00·17-s − 1.33·19-s − 0.218·21-s + 0.208·23-s − 0.994·25-s + 0.192·27-s + 0.928·29-s − 0.0351·31-s + 0.314·33-s + 0.0283·35-s − 0.548·37-s − 0.582·39-s − 0.772·41-s − 0.198·43-s − 0.0249·45-s + 0.457·47-s + 0.142·49-s + 0.579·51-s + 0.454·53-s − 0.0407·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 0.167T + 5T^{2} \) |
| 11 | \( 1 - 1.80T + 11T^{2} \) |
| 13 | \( 1 + 3.63T + 13T^{2} \) |
| 17 | \( 1 - 4.13T + 17T^{2} \) |
| 19 | \( 1 + 5.80T + 19T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 0.195T + 31T^{2} \) |
| 37 | \( 1 + 3.33T + 37T^{2} \) |
| 41 | \( 1 + 4.94T + 41T^{2} \) |
| 43 | \( 1 + 1.30T + 43T^{2} \) |
| 47 | \( 1 - 3.13T + 47T^{2} \) |
| 53 | \( 1 - 3.30T + 53T^{2} \) |
| 59 | \( 1 - 4.91T + 59T^{2} \) |
| 61 | \( 1 - 8.77T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 8.77T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 1.52T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 - 8.77T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−7.44707178851839721215389747583, −6.97506283726966712610440506470, −6.19377945286188496315444339444, −5.44128282280585246691598326272, −4.52050624535094238869098194623, −3.91360949497471905638325146021, −3.07902687060187516817010928700, −2.33770775849185224575453048889, −1.38532754450748156164289082179, 0,
1.38532754450748156164289082179, 2.33770775849185224575453048889, 3.07902687060187516817010928700, 3.91360949497471905638325146021, 4.52050624535094238869098194623, 5.44128282280585246691598326272, 6.19377945286188496315444339444, 6.97506283726966712610440506470, 7.44707178851839721215389747583
