| L(s) = 1 | + 3-s + 4.14·5-s + 1.03·7-s + 9-s − 3.15·11-s − 4.46·13-s + 4.14·15-s − 7.46·17-s − 4.84·19-s + 1.03·21-s − 5.72·23-s + 12.1·25-s + 27-s + 4.37·29-s − 11.0·31-s − 3.15·33-s + 4.29·35-s + 1.22·37-s − 4.46·39-s − 2.29·41-s − 0.00419·43-s + 4.14·45-s − 5.10·47-s − 5.92·49-s − 7.46·51-s + 5.05·53-s − 13.0·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.85·5-s + 0.391·7-s + 0.333·9-s − 0.950·11-s − 1.23·13-s + 1.06·15-s − 1.81·17-s − 1.11·19-s + 0.226·21-s − 1.19·23-s + 2.43·25-s + 0.192·27-s + 0.813·29-s − 1.99·31-s − 0.548·33-s + 0.726·35-s + 0.201·37-s − 0.715·39-s − 0.358·41-s − 0.000639·43-s + 0.617·45-s − 0.745·47-s − 0.846·49-s − 1.04·51-s + 0.694·53-s − 1.76·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 4.14T + 5T^{2} \) |
| 7 | \( 1 - 1.03T + 7T^{2} \) |
| 11 | \( 1 + 3.15T + 11T^{2} \) |
| 13 | \( 1 + 4.46T + 13T^{2} \) |
| 17 | \( 1 + 7.46T + 17T^{2} \) |
| 19 | \( 1 + 4.84T + 19T^{2} \) |
| 23 | \( 1 + 5.72T + 23T^{2} \) |
| 29 | \( 1 - 4.37T + 29T^{2} \) |
| 31 | \( 1 + 11.0T + 31T^{2} \) |
| 37 | \( 1 - 1.22T + 37T^{2} \) |
| 41 | \( 1 + 2.29T + 41T^{2} \) |
| 43 | \( 1 + 0.00419T + 43T^{2} \) |
| 47 | \( 1 + 5.10T + 47T^{2} \) |
| 53 | \( 1 - 5.05T + 53T^{2} \) |
| 59 | \( 1 + 8.22T + 59T^{2} \) |
| 61 | \( 1 - 2.02T + 61T^{2} \) |
| 67 | \( 1 - 2.70T + 67T^{2} \) |
| 71 | \( 1 - 3.08T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 - 3.69T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 - 2.58T + 97T^{2} \) |
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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
−7.53733737614750217612441553142, −6.59664820060408939202330697773, −6.29782053298092533446402080733, −5.14645633159901732860747258925, −4.97377559757926615549853412800, −3.95181254586925686845297732236, −2.64912829251183526818614912466, −2.20544410073396800348871648139, −1.78798190345569241025000424516, 0,
1.78798190345569241025000424516, 2.20544410073396800348871648139, 2.64912829251183526818614912466, 3.95181254586925686845297732236, 4.97377559757926615549853412800, 5.14645633159901732860747258925, 6.29782053298092533446402080733, 6.59664820060408939202330697773, 7.53733737614750217612441553142
