| L(s) = 1 | + 1.27·3-s + 3.38·5-s − 1.38·9-s + 1.41·11-s − 2.65·13-s + 4.30·15-s − 5.44·17-s − 2.96·19-s − 23-s + 6.43·25-s − 5.57·27-s + 1.24·29-s − 3.98·31-s + 1.79·33-s − 7.95·37-s − 3.37·39-s − 2.42·41-s − 9.02·43-s − 4.67·45-s + 10.4·47-s − 6.91·51-s + 1.42·53-s + 4.77·55-s − 3.76·57-s − 14.5·59-s − 11.4·61-s − 8.97·65-s + ⋯ |
| L(s) = 1 | + 0.734·3-s + 1.51·5-s − 0.460·9-s + 0.425·11-s − 0.735·13-s + 1.11·15-s − 1.31·17-s − 0.679·19-s − 0.208·23-s + 1.28·25-s − 1.07·27-s + 0.230·29-s − 0.716·31-s + 0.312·33-s − 1.30·37-s − 0.540·39-s − 0.378·41-s − 1.37·43-s − 0.697·45-s + 1.52·47-s − 0.968·51-s + 0.195·53-s + 0.644·55-s − 0.499·57-s − 1.89·59-s − 1.46·61-s − 1.11·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 1.27T + 3T^{2} \) |
| 5 | \( 1 - 3.38T + 5T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 + 2.65T + 13T^{2} \) |
| 17 | \( 1 + 5.44T + 17T^{2} \) |
| 19 | \( 1 + 2.96T + 19T^{2} \) |
| 29 | \( 1 - 1.24T + 29T^{2} \) |
| 31 | \( 1 + 3.98T + 31T^{2} \) |
| 37 | \( 1 + 7.95T + 37T^{2} \) |
| 41 | \( 1 + 2.42T + 41T^{2} \) |
| 43 | \( 1 + 9.02T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 1.42T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 6.95T + 67T^{2} \) |
| 71 | \( 1 - 5.52T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 - 5.28T + 79T^{2} \) |
| 83 | \( 1 + 6.62T + 83T^{2} \) |
| 89 | \( 1 - 9.70T + 89T^{2} \) |
| 97 | \( 1 - 4.19T + 97T^{2} \) |
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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
−7.36145438562235925111052655251, −6.58689323444263543772919178697, −6.14372199663971143157762151856, −5.33231935013788178727351289720, −4.69799957984220703656039518896, −3.73197693391420733029565051616, −2.84175371945746533820007338565, −2.14576549724311118925661264886, −1.69122095261105226766520999103, 0,
1.69122095261105226766520999103, 2.14576549724311118925661264886, 2.84175371945746533820007338565, 3.73197693391420733029565051616, 4.69799957984220703656039518896, 5.33231935013788178727351289720, 6.14372199663971143157762151856, 6.58689323444263543772919178697, 7.36145438562235925111052655251
