| L(s) = 1 | + 4·3-s − 8·4-s − 5·5-s − 22·7-s − 11·9-s − 12·11-s − 32·12-s + 8·13-s − 20·15-s + 64·16-s − 66·17-s + 19·19-s + 40·20-s − 88·21-s − 30·23-s + 25·25-s − 152·27-s + 176·28-s − 6·29-s − 64·31-s − 48·33-s + 110·35-s + 88·36-s − 16·37-s + 32·39-s + 54·41-s + 182·43-s + ⋯ |
| L(s) = 1 | + 0.769·3-s − 4-s − 0.447·5-s − 1.18·7-s − 0.407·9-s − 0.328·11-s − 0.769·12-s + 0.170·13-s − 0.344·15-s + 16-s − 0.941·17-s + 0.229·19-s + 0.447·20-s − 0.914·21-s − 0.271·23-s + 1/5·25-s − 1.08·27-s + 1.18·28-s − 0.0384·29-s − 0.370·31-s − 0.253·33-s + 0.531·35-s + 0.407·36-s − 0.0710·37-s + 0.131·39-s + 0.205·41-s + 0.645·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + p T \) |
| 19 | \( 1 - p T \) |
| good | 2 | \( 1 + p^{3} T^{2} \) |
| 3 | \( 1 - 4 T + p^{3} T^{2} \) |
| 7 | \( 1 + 22 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 8 T + p^{3} T^{2} \) |
| 17 | \( 1 + 66 T + p^{3} T^{2} \) |
| 23 | \( 1 + 30 T + p^{3} T^{2} \) |
| 29 | \( 1 + 6 T + p^{3} T^{2} \) |
| 31 | \( 1 + 64 T + p^{3} T^{2} \) |
| 37 | \( 1 + 16 T + p^{3} T^{2} \) |
| 41 | \( 1 - 54 T + p^{3} T^{2} \) |
| 43 | \( 1 - 182 T + p^{3} T^{2} \) |
| 47 | \( 1 - 594 T + p^{3} T^{2} \) |
| 53 | \( 1 - 396 T + p^{3} T^{2} \) |
| 59 | \( 1 + 564 T + p^{3} T^{2} \) |
| 61 | \( 1 + 706 T + p^{3} T^{2} \) |
| 67 | \( 1 + 628 T + p^{3} T^{2} \) |
| 71 | \( 1 + 984 T + p^{3} T^{2} \) |
| 73 | \( 1 - 14 T + p^{3} T^{2} \) |
| 79 | \( 1 + 328 T + p^{3} T^{2} \) |
| 83 | \( 1 + 294 T + p^{3} T^{2} \) |
| 89 | \( 1 - 918 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1564 T + p^{3} T^{2} \) |
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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
−13.23176400295993136278137173809, −12.18529288226934930129796606833, −10.61861488346714917671902048705, −9.349203326536036221441078884849, −8.741617711868389799658203617073, −7.52398933923643560599920672654, −5.87023200879182968816213133068, −4.16314588786257167622208555575, −2.98001475540282081458606345842, 0,
2.98001475540282081458606345842, 4.16314588786257167622208555575, 5.87023200879182968816213133068, 7.52398933923643560599920672654, 8.741617711868389799658203617073, 9.349203326536036221441078884849, 10.61861488346714917671902048705, 12.18529288226934930129796606833, 13.23176400295993136278137173809
