| L(s) = 1 | + 2-s + 3-s + 4-s + 1.51·5-s + 6-s − 2.59·7-s + 8-s + 9-s + 1.51·10-s − 5.00·11-s + 12-s − 4.82·13-s − 2.59·14-s + 1.51·15-s + 16-s − 0.863·17-s + 18-s − 7.00·19-s + 1.51·20-s − 2.59·21-s − 5.00·22-s + 24-s − 2.70·25-s − 4.82·26-s + 27-s − 2.59·28-s + 3.11·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.676·5-s + 0.408·6-s − 0.980·7-s + 0.353·8-s + 0.333·9-s + 0.478·10-s − 1.50·11-s + 0.288·12-s − 1.33·13-s − 0.693·14-s + 0.390·15-s + 0.250·16-s − 0.209·17-s + 0.235·18-s − 1.60·19-s + 0.338·20-s − 0.566·21-s − 1.06·22-s + 0.204·24-s − 0.541·25-s − 0.945·26-s + 0.192·27-s − 0.490·28-s + 0.578·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 1.51T + 5T^{2} \) |
| 7 | \( 1 + 2.59T + 7T^{2} \) |
| 11 | \( 1 + 5.00T + 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 + 0.863T + 17T^{2} \) |
| 19 | \( 1 + 7.00T + 19T^{2} \) |
| 29 | \( 1 - 3.11T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 + 0.602T + 37T^{2} \) |
| 41 | \( 1 - 5.29T + 41T^{2} \) |
| 43 | \( 1 - 1.45T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 - 4.23T + 53T^{2} \) |
| 59 | \( 1 - 3.98T + 59T^{2} \) |
| 61 | \( 1 + 7.26T + 61T^{2} \) |
| 67 | \( 1 + 16.1T + 67T^{2} \) |
| 71 | \( 1 - 5.88T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 2.64T + 83T^{2} \) |
| 89 | \( 1 + 5.23T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−8.157601299192614158011600216897, −7.52959709178523827539422599729, −6.64848333306843583732254730990, −6.06576843839443105086646227454, −5.13442929690058577344620375544, −4.48760839683434983360374068940, −3.38686753021798801067647457681, −2.53374920360385843472521063646, −2.10855052942217064717028053009, 0,
2.10855052942217064717028053009, 2.53374920360385843472521063646, 3.38686753021798801067647457681, 4.48760839683434983360374068940, 5.13442929690058577344620375544, 6.06576843839443105086646227454, 6.64848333306843583732254730990, 7.52959709178523827539422599729, 8.157601299192614158011600216897
