L(s) = 1 | + 2-s − 3.11·3-s + 4-s + 0.453·5-s − 3.11·6-s − 1.01·7-s + 8-s + 6.72·9-s + 0.453·10-s + 6.08·11-s − 3.11·12-s − 1.01·14-s − 1.41·15-s + 16-s − 2.16·17-s + 6.72·18-s + 19-s + 0.453·20-s + 3.16·21-s + 6.08·22-s − 7.53·23-s − 3.11·24-s − 4.79·25-s − 11.6·27-s − 1.01·28-s − 5.73·29-s − 1.41·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.80·3-s + 0.5·4-s + 0.202·5-s − 1.27·6-s − 0.383·7-s + 0.353·8-s + 2.24·9-s + 0.143·10-s + 1.83·11-s − 0.900·12-s − 0.271·14-s − 0.364·15-s + 0.250·16-s − 0.525·17-s + 1.58·18-s + 0.229·19-s + 0.101·20-s + 0.690·21-s + 1.29·22-s − 1.57·23-s − 0.636·24-s − 0.958·25-s − 2.23·27-s − 0.191·28-s − 1.06·29-s − 0.257·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + 3.11T + 3T^{2} \) |
| 5 | \( 1 - 0.453T + 5T^{2} \) |
| 7 | \( 1 + 1.01T + 7T^{2} \) |
| 11 | \( 1 - 6.08T + 11T^{2} \) |
| 17 | \( 1 + 2.16T + 17T^{2} \) |
| 23 | \( 1 + 7.53T + 23T^{2} \) |
| 29 | \( 1 + 5.73T + 29T^{2} \) |
| 31 | \( 1 - 0.824T + 31T^{2} \) |
| 37 | \( 1 + 2.70T + 37T^{2} \) |
| 41 | \( 1 - 1.79T + 41T^{2} \) |
| 43 | \( 1 + 4.64T + 43T^{2} \) |
| 47 | \( 1 - 2.03T + 47T^{2} \) |
| 53 | \( 1 - 1.29T + 53T^{2} \) |
| 59 | \( 1 - 2.64T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 - 5.97T + 89T^{2} \) |
| 97 | \( 1 - 9.06T + 97T^{2} \) |
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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
−7.16801405486205687760044856862, −6.68435552049258914859221300104, −6.12189590823504330405382843374, −5.70639712902510946279713237867, −4.90172458435927267474752754453, −4.00108036113193366024380309295, −3.75955482373927900500690237539, −2.07551247884405661061896706788, −1.29072487846207445617789282962, 0,
1.29072487846207445617789282962, 2.07551247884405661061896706788, 3.75955482373927900500690237539, 4.00108036113193366024380309295, 4.90172458435927267474752754453, 5.70639712902510946279713237867, 6.12189590823504330405382843374, 6.68435552049258914859221300104, 7.16801405486205687760044856862