| L(s) = 1 | + 3·3-s + 10.6·5-s − 7·7-s + 9·9-s − 68.4·11-s + 23.0·13-s + 31.8·15-s − 62.2·17-s − 53.2·19-s − 21·21-s − 89.8·23-s − 11.9·25-s + 27·27-s − 43.2·29-s − 102.·31-s − 205.·33-s − 74.4·35-s − 302.·37-s + 69.1·39-s + 73.4·41-s + 377.·43-s + 95.6·45-s − 487.·47-s + 49·49-s − 186.·51-s + 467.·53-s − 727.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.951·5-s − 0.377·7-s + 0.333·9-s − 1.87·11-s + 0.492·13-s + 0.549·15-s − 0.887·17-s − 0.643·19-s − 0.218·21-s − 0.815·23-s − 0.0954·25-s + 0.192·27-s − 0.277·29-s − 0.595·31-s − 1.08·33-s − 0.359·35-s − 1.34·37-s + 0.284·39-s + 0.279·41-s + 1.33·43-s + 0.317·45-s − 1.51·47-s + 0.142·49-s − 0.512·51-s + 1.21·53-s − 1.78·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| good | 5 | \( 1 - 10.6T + 125T^{2} \) |
| 11 | \( 1 + 68.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 23.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 62.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 53.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 89.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 43.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 102.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 302.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 73.4T + 6.89e4T^{2} \) |
| 43 | \( 1 - 377.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 487.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 467.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 432.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 70.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 475.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 680.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 604.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 329.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 834.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 947.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 661.T + 9.12e5T^{2} \) |
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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
−9.745361015780953176444784491813, −8.797810610596652073914812714147, −8.063375487041641279414039284218, −7.04527242191358544064849249382, −6.00775052306050073352414982563, −5.21878200032402990488802631761, −3.93832880410780802369591439489, −2.65685688973193939387331508531, −1.93308072396651828791255861111, 0,
1.93308072396651828791255861111, 2.65685688973193939387331508531, 3.93832880410780802369591439489, 5.21878200032402990488802631761, 6.00775052306050073352414982563, 7.04527242191358544064849249382, 8.063375487041641279414039284218, 8.797810610596652073914812714147, 9.745361015780953176444784491813
