| L(s) = 1 | + 1.61·3-s − 0.618·7-s − 0.381·9-s + 2.85·11-s + 7.09·13-s − 6.09·17-s − 1.85·19-s − 1.00·21-s + 23-s − 5.47·27-s − 9.23·29-s − 9.09·31-s + 4.61·33-s − 6.47·37-s + 11.4·39-s + 3.32·41-s − 3.70·47-s − 6.61·49-s − 9.85·51-s − 0.472·53-s − 3·57-s − 1.70·59-s − 9.32·61-s + 0.236·63-s + 14.4·67-s + 1.61·69-s + 4.09·71-s + ⋯ |
| L(s) = 1 | + 0.934·3-s − 0.233·7-s − 0.127·9-s + 0.860·11-s + 1.96·13-s − 1.47·17-s − 0.425·19-s − 0.218·21-s + 0.208·23-s − 1.05·27-s − 1.71·29-s − 1.63·31-s + 0.803·33-s − 1.06·37-s + 1.83·39-s + 0.519·41-s − 0.540·47-s − 0.945·49-s − 1.37·51-s − 0.0648·53-s − 0.397·57-s − 0.222·59-s − 1.19·61-s + 0.0297·63-s + 1.76·67-s + 0.194·69-s + 0.485·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 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
−7.40177112914968443631161622888, −6.69397454932319903239494361539, −6.10984777838260858791294886957, −5.43858820748899125312616633818, −4.26413467308948680632130394897, −3.70079567952385446561710356643, −3.25523281088490358691779076833, −2.08697807317832002030167907154, −1.54015812469677655808674700476, 0,
1.54015812469677655808674700476, 2.08697807317832002030167907154, 3.25523281088490358691779076833, 3.70079567952385446561710356643, 4.26413467308948680632130394897, 5.43858820748899125312616633818, 6.10984777838260858791294886957, 6.69397454932319903239494361539, 7.40177112914968443631161622888
