| L(s) = 1 | − 3-s − 3.56·5-s − 0.561·7-s + 9-s + 2·11-s + 3.56·15-s − 1.56·17-s − 7.12·19-s + 0.561·21-s − 2·23-s + 7.68·25-s − 27-s + 6.68·29-s − 2.56·31-s − 2·33-s + 2·35-s + 7.56·37-s − 1.56·41-s − 4.56·43-s − 3.56·45-s − 8.24·47-s − 6.68·49-s + 1.56·51-s − 0.684·53-s − 7.12·55-s + 7.12·57-s + 2.87·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.59·5-s − 0.212·7-s + 0.333·9-s + 0.603·11-s + 0.919·15-s − 0.378·17-s − 1.63·19-s + 0.122·21-s − 0.417·23-s + 1.53·25-s − 0.192·27-s + 1.24·29-s − 0.460·31-s − 0.348·33-s + 0.338·35-s + 1.24·37-s − 0.243·41-s − 0.695·43-s − 0.530·45-s − 1.20·47-s − 0.954·49-s + 0.218·51-s − 0.0940·53-s − 0.960·55-s + 0.943·57-s + 0.374·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5173804238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5173804238\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 3.56T + 5T^{2} \) |
| 7 | \( 1 + 0.561T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 + 1.56T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 + 2.56T + 31T^{2} \) |
| 37 | \( 1 - 7.56T + 37T^{2} \) |
| 41 | \( 1 + 1.56T + 41T^{2} \) |
| 43 | \( 1 + 4.56T + 43T^{2} \) |
| 47 | \( 1 + 8.24T + 47T^{2} \) |
| 53 | \( 1 + 0.684T + 53T^{2} \) |
| 59 | \( 1 - 2.87T + 59T^{2} \) |
| 61 | \( 1 - 3.87T + 61T^{2} \) |
| 67 | \( 1 + 4.56T + 67T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 5.43T + 79T^{2} \) |
| 83 | \( 1 - 0.876T + 83T^{2} \) |
| 89 | \( 1 - 4.87T + 89T^{2} \) |
| 97 | \( 1 + 8.56T + 97T^{2} \) |
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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
−7.896800320392772036708691283164, −6.99723379377983517710511435693, −6.56748112265990479895797823892, −5.90052699246625139778732580003, −4.69192884917989858401857642703, −4.38466829270105116936186900874, −3.69234183090953880527652852295, −2.82706628027420485498912758518, −1.59431596078082280046255954366, −0.36867712249384187583558968223,
0.36867712249384187583558968223, 1.59431596078082280046255954366, 2.82706628027420485498912758518, 3.69234183090953880527652852295, 4.38466829270105116936186900874, 4.69192884917989858401857642703, 5.90052699246625139778732580003, 6.56748112265990479895797823892, 6.99723379377983517710511435693, 7.896800320392772036708691283164
