L(s) = 1 | + 2-s − 3-s + 4-s + 1.44·5-s − 6-s + 0.367·7-s + 8-s + 9-s + 1.44·10-s + 0.547·11-s − 12-s + 13-s + 0.367·14-s − 1.44·15-s + 16-s − 3.55·17-s + 18-s − 4.08·19-s + 1.44·20-s − 0.367·21-s + 0.547·22-s − 6.44·23-s − 24-s − 2.91·25-s + 26-s − 27-s + 0.367·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.645·5-s − 0.408·6-s + 0.138·7-s + 0.353·8-s + 0.333·9-s + 0.456·10-s + 0.165·11-s − 0.288·12-s + 0.277·13-s + 0.0982·14-s − 0.372·15-s + 0.250·16-s − 0.862·17-s + 0.235·18-s − 0.937·19-s + 0.322·20-s − 0.0802·21-s + 0.116·22-s − 1.34·23-s − 0.204·24-s − 0.583·25-s + 0.196·26-s − 0.192·27-s + 0.0694·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 - 1.44T + 5T^{2} \) |
| 7 | \( 1 - 0.367T + 7T^{2} \) |
| 11 | \( 1 - 0.547T + 11T^{2} \) |
| 17 | \( 1 + 3.55T + 17T^{2} \) |
| 19 | \( 1 + 4.08T + 19T^{2} \) |
| 23 | \( 1 + 6.44T + 23T^{2} \) |
| 29 | \( 1 - 0.496T + 29T^{2} \) |
| 31 | \( 1 - 1.35T + 31T^{2} \) |
| 37 | \( 1 + 1.55T + 37T^{2} \) |
| 41 | \( 1 + 0.751T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 7.51T + 47T^{2} \) |
| 53 | \( 1 - 1.23T + 53T^{2} \) |
| 59 | \( 1 - 3.24T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 7.76T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 8.69T + 83T^{2} \) |
| 89 | \( 1 + 3.90T + 89T^{2} \) |
| 97 | \( 1 - 0.761T + 97T^{2} \) |
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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
−7.30730269250499233037839165059, −6.46448330562562726806122354526, −6.06424201476245632081038464571, −5.56405415740169791482018182202, −4.44869670014349503845229272338, −4.29561490028019678161597797105, −3.14778001069062404703597110003, −2.14477704181566443501137935634, −1.54738789717763171414539532389, 0,
1.54738789717763171414539532389, 2.14477704181566443501137935634, 3.14778001069062404703597110003, 4.29561490028019678161597797105, 4.44869670014349503845229272338, 5.56405415740169791482018182202, 6.06424201476245632081038464571, 6.46448330562562726806122354526, 7.30730269250499233037839165059