| L(s) = 1 | + 5-s + 1.43·7-s − 1.35·11-s − 5.52·13-s + 4.82·17-s − 0.648·19-s − 8.90·23-s + 25-s − 7.17·29-s − 4.64·31-s + 1.43·35-s + 1.35·37-s + 0.351·41-s + 4.82·43-s − 9.49·47-s − 4.93·49-s − 8.17·53-s − 1.35·55-s + 1.46·59-s + 6.69·61-s − 5.52·65-s + 12.4·67-s + 2.22·71-s − 4.34·73-s − 1.94·77-s − 13.0·79-s − 5.26·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.543·7-s − 0.407·11-s − 1.53·13-s + 1.16·17-s − 0.148·19-s − 1.85·23-s + 0.200·25-s − 1.33·29-s − 0.834·31-s + 0.243·35-s + 0.222·37-s + 0.0549·41-s + 0.735·43-s − 1.38·47-s − 0.704·49-s − 1.12·53-s − 0.182·55-s + 0.191·59-s + 0.857·61-s − 0.685·65-s + 1.51·67-s + 0.264·71-s − 0.508·73-s − 0.221·77-s − 1.46·79-s − 0.577·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 - 1.43T + 7T^{2} \) |
| 11 | \( 1 + 1.35T + 11T^{2} \) |
| 13 | \( 1 + 5.52T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 + 0.648T + 19T^{2} \) |
| 23 | \( 1 + 8.90T + 23T^{2} \) |
| 29 | \( 1 + 7.17T + 29T^{2} \) |
| 31 | \( 1 + 4.64T + 31T^{2} \) |
| 37 | \( 1 - 1.35T + 37T^{2} \) |
| 41 | \( 1 - 0.351T + 41T^{2} \) |
| 43 | \( 1 - 4.82T + 43T^{2} \) |
| 47 | \( 1 + 9.49T + 47T^{2} \) |
| 53 | \( 1 + 8.17T + 53T^{2} \) |
| 59 | \( 1 - 1.46T + 59T^{2} \) |
| 61 | \( 1 - 6.69T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 2.22T + 71T^{2} \) |
| 73 | \( 1 + 4.34T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + 5.26T + 83T^{2} \) |
| 89 | \( 1 + 11T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−7.984889893692901635694770787147, −7.76306359163137921394737133285, −6.86485781274083215863915481504, −5.80677020327984417436825529242, −5.32757101976363908454209091488, −4.49404006686436724393699435076, −3.50078400006305686366418218647, −2.39964634950919288564449237155, −1.66813549539373760459310320579, 0,
1.66813549539373760459310320579, 2.39964634950919288564449237155, 3.50078400006305686366418218647, 4.49404006686436724393699435076, 5.32757101976363908454209091488, 5.80677020327984417436825529242, 6.86485781274083215863915481504, 7.76306359163137921394737133285, 7.984889893692901635694770787147
