| L(s) = 1 | − 0.561·5-s − 7-s + 5.12·11-s + 13-s − 7.12·17-s + 2.56·19-s + 3.68·23-s − 4.68·25-s − 4.56·29-s − 1.43·31-s + 0.561·35-s − 7.12·37-s + 2·41-s − 5.43·43-s − 5.43·47-s + 49-s + 5.68·53-s − 2.87·55-s + 8·59-s − 2·61-s − 0.561·65-s + 1.12·67-s − 6.24·71-s − 12.5·73-s − 5.12·77-s − 3.68·79-s + 14.5·83-s + ⋯ |
| L(s) = 1 | − 0.251·5-s − 0.377·7-s + 1.54·11-s + 0.277·13-s − 1.72·17-s + 0.587·19-s + 0.768·23-s − 0.936·25-s − 0.847·29-s − 0.258·31-s + 0.0949·35-s − 1.17·37-s + 0.312·41-s − 0.829·43-s − 0.793·47-s + 0.142·49-s + 0.780·53-s − 0.387·55-s + 1.04·59-s − 0.256·61-s − 0.0696·65-s + 0.137·67-s − 0.741·71-s − 1.47·73-s − 0.583·77-s − 0.414·79-s + 1.59·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 0.561T + 5T^{2} \) |
| 11 | \( 1 - 5.12T + 11T^{2} \) |
| 17 | \( 1 + 7.12T + 17T^{2} \) |
| 19 | \( 1 - 2.56T + 19T^{2} \) |
| 23 | \( 1 - 3.68T + 23T^{2} \) |
| 29 | \( 1 + 4.56T + 29T^{2} \) |
| 31 | \( 1 + 1.43T + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 5.43T + 43T^{2} \) |
| 47 | \( 1 + 5.43T + 47T^{2} \) |
| 53 | \( 1 - 5.68T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 1.12T + 67T^{2} \) |
| 71 | \( 1 + 6.24T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 + 3.68T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 - 5.68T + 97T^{2} \) |
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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
−7.52107256521758958207755911254, −6.85404975948450950797668240898, −6.42916898177105447712614891805, −5.58115999147749832034796371750, −4.67829370240888437049142254285, −3.89321305068040487314887289949, −3.41058722906170967680983244591, −2.21922679133019128242001569332, −1.33263030637445648412389089078, 0,
1.33263030637445648412389089078, 2.21922679133019128242001569332, 3.41058722906170967680983244591, 3.89321305068040487314887289949, 4.67829370240888437049142254285, 5.58115999147749832034796371750, 6.42916898177105447712614891805, 6.85404975948450950797668240898, 7.52107256521758958207755911254
