| L(s) = 1 | + 2-s + 4-s − 1.63·5-s + 4.18·7-s + 8-s − 1.63·10-s − 0.197·11-s − 5.53·13-s + 4.18·14-s + 16-s + 6.24·17-s − 1.76·19-s − 1.63·20-s − 0.197·22-s − 1.25·23-s − 2.32·25-s − 5.53·26-s + 4.18·28-s − 6.46·29-s − 9.67·31-s + 32-s + 6.24·34-s − 6.83·35-s − 8.61·37-s − 1.76·38-s − 1.63·40-s − 7.00·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.731·5-s + 1.58·7-s + 0.353·8-s − 0.517·10-s − 0.0594·11-s − 1.53·13-s + 1.11·14-s + 0.250·16-s + 1.51·17-s − 0.404·19-s − 0.365·20-s − 0.0420·22-s − 0.262·23-s − 0.465·25-s − 1.08·26-s + 0.790·28-s − 1.20·29-s − 1.73·31-s + 0.176·32-s + 1.07·34-s − 1.15·35-s − 1.41·37-s − 0.286·38-s − 0.258·40-s − 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 + 1.63T + 5T^{2} \) |
| 7 | \( 1 - 4.18T + 7T^{2} \) |
| 11 | \( 1 + 0.197T + 11T^{2} \) |
| 13 | \( 1 + 5.53T + 13T^{2} \) |
| 17 | \( 1 - 6.24T + 17T^{2} \) |
| 19 | \( 1 + 1.76T + 19T^{2} \) |
| 23 | \( 1 + 1.25T + 23T^{2} \) |
| 29 | \( 1 + 6.46T + 29T^{2} \) |
| 31 | \( 1 + 9.67T + 31T^{2} \) |
| 37 | \( 1 + 8.61T + 37T^{2} \) |
| 41 | \( 1 + 7.00T + 41T^{2} \) |
| 43 | \( 1 - 4.69T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 5.96T + 53T^{2} \) |
| 59 | \( 1 - 4.41T + 59T^{2} \) |
| 61 | \( 1 - 6.34T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 - 8.52T + 71T^{2} \) |
| 73 | \( 1 + 0.842T + 73T^{2} \) |
| 79 | \( 1 + 2.79T + 79T^{2} \) |
| 83 | \( 1 - 1.68T + 83T^{2} \) |
| 89 | \( 1 - 0.518T + 89T^{2} \) |
| 97 | \( 1 - 2.99T + 97T^{2} \) |
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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
−7.53160707018470867042666365844, −7.03976561449997268162130678534, −5.79339093671433242521351454993, −5.21753580964308832604672570207, −4.79913079590025838739779057975, −3.91613459647157127797373669635, −3.34415385522384172646232664811, −2.15006443792993029971209988832, −1.58352882615974936117732675492, 0,
1.58352882615974936117732675492, 2.15006443792993029971209988832, 3.34415385522384172646232664811, 3.91613459647157127797373669635, 4.79913079590025838739779057975, 5.21753580964308832604672570207, 5.79339093671433242521351454993, 7.03976561449997268162130678534, 7.53160707018470867042666365844
