| L(s) = 1 | − 1.61·3-s + 5-s + 0.618·7-s − 0.381·9-s + 2.85·11-s − 7.09·13-s − 1.61·15-s + 6.09·17-s − 1.85·19-s − 1.00·21-s − 23-s + 25-s + 5.47·27-s − 9.23·29-s − 9.09·31-s − 4.61·33-s + 0.618·35-s + 6.47·37-s + 11.4·39-s + 3.32·41-s − 0.381·45-s + 3.70·47-s − 6.61·49-s − 9.85·51-s + 0.472·53-s + 2.85·55-s + 3·57-s + ⋯ |
| L(s) = 1 | − 0.934·3-s + 0.447·5-s + 0.233·7-s − 0.127·9-s + 0.860·11-s − 1.96·13-s − 0.417·15-s + 1.47·17-s − 0.425·19-s − 0.218·21-s − 0.208·23-s + 0.200·25-s + 1.05·27-s − 1.71·29-s − 1.63·31-s − 0.803·33-s + 0.104·35-s + 1.06·37-s + 1.83·39-s + 0.519·41-s − 0.0569·45-s + 0.540·47-s − 0.945·49-s − 1.37·51-s + 0.0648·53-s + 0.384·55-s + 0.397·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 1.61T + 3T^{2} \) |
| 7 | \( 1 - 0.618T + 7T^{2} \) |
| 11 | \( 1 - 2.85T + 11T^{2} \) |
| 13 | \( 1 + 7.09T + 13T^{2} \) |
| 17 | \( 1 - 6.09T + 17T^{2} \) |
| 19 | \( 1 + 1.85T + 19T^{2} \) |
| 29 | \( 1 + 9.23T + 29T^{2} \) |
| 31 | \( 1 + 9.09T + 31T^{2} \) |
| 37 | \( 1 - 6.47T + 37T^{2} \) |
| 41 | \( 1 - 3.32T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 3.70T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 + 1.70T + 59T^{2} \) |
| 61 | \( 1 + 9.32T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 - 4.09T + 71T^{2} \) |
| 73 | \( 1 - 3.23T + 73T^{2} \) |
| 79 | \( 1 + 1.52T + 79T^{2} \) |
| 83 | \( 1 - 6.94T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−9.192606943682537249560077900761, −7.82777677525183601533590119977, −7.30237624592909503089433211277, −6.29585942859912162290880847594, −5.57439464092130833527045222897, −5.03172832639205616633759214658, −3.96757502714772256063808846325, −2.71591056579124554536689308591, −1.53776077713781743186760966496, 0,
1.53776077713781743186760966496, 2.71591056579124554536689308591, 3.96757502714772256063808846325, 5.03172832639205616633759214658, 5.57439464092130833527045222897, 6.29585942859912162290880847594, 7.30237624592909503089433211277, 7.82777677525183601533590119977, 9.192606943682537249560077900761
