| L(s) = 1 | + 2-s + 3-s + 4-s − 4.24·5-s + 6-s + 7-s + 8-s + 9-s − 4.24·10-s − 0.198·11-s + 12-s + 14-s − 4.24·15-s + 16-s − 0.664·17-s + 18-s − 2.91·19-s − 4.24·20-s + 21-s − 0.198·22-s + 1.96·23-s + 24-s + 13.0·25-s + 27-s + 28-s + 4.65·29-s − 4.24·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.89·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 1.34·10-s − 0.0597·11-s + 0.288·12-s + 0.267·14-s − 1.09·15-s + 0.250·16-s − 0.161·17-s + 0.235·18-s − 0.668·19-s − 0.949·20-s + 0.218·21-s − 0.0422·22-s + 0.408·23-s + 0.204·24-s + 2.60·25-s + 0.192·27-s + 0.188·28-s + 0.864·29-s − 0.775·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 4.24T + 5T^{2} \) |
| 11 | \( 1 + 0.198T + 11T^{2} \) |
| 17 | \( 1 + 0.664T + 17T^{2} \) |
| 19 | \( 1 + 2.91T + 19T^{2} \) |
| 23 | \( 1 - 1.96T + 23T^{2} \) |
| 29 | \( 1 - 4.65T + 29T^{2} \) |
| 31 | \( 1 + 6.67T + 31T^{2} \) |
| 37 | \( 1 - 4.96T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 2.85T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 14.5T + 53T^{2} \) |
| 59 | \( 1 - 5.05T + 59T^{2} \) |
| 61 | \( 1 + 0.878T + 61T^{2} \) |
| 67 | \( 1 - 6.03T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 + 4.84T + 73T^{2} \) |
| 79 | \( 1 - 0.423T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 + 1.47T + 97T^{2} \) |
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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
−7.64134299437464353791132133882, −6.93591323045714733440255940818, −6.40041484120188797609936776030, −5.02036399523995333640704698599, −4.71730452925961442318067791754, −3.83231141584474201706806802508, −3.42366586246942083503164432799, −2.57479431814433403163452977269, −1.40701266521261059854509410140, 0,
1.40701266521261059854509410140, 2.57479431814433403163452977269, 3.42366586246942083503164432799, 3.83231141584474201706806802508, 4.71730452925961442318067791754, 5.02036399523995333640704698599, 6.40041484120188797609936776030, 6.93591323045714733440255940818, 7.64134299437464353791132133882
