| L(s) = 1 | − 1.61·3-s − 0.618·7-s − 0.381·9-s + 11-s − 2.23·13-s − 4.85·17-s + 5.47·19-s + 1.00·21-s + 6.32·23-s + 5.47·27-s − 4.38·29-s + 4.23·31-s − 1.61·33-s − 1.76·37-s + 3.61·39-s + 7.94·41-s − 8.47·43-s − 4.70·47-s − 6.61·49-s + 7.85·51-s + 3.85·53-s − 8.85·57-s + 3.76·59-s + 7.09·61-s + 0.236·63-s − 10.2·69-s + 0.291·71-s + ⋯ |
| L(s) = 1 | − 0.934·3-s − 0.233·7-s − 0.127·9-s + 0.301·11-s − 0.620·13-s − 1.17·17-s + 1.25·19-s + 0.218·21-s + 1.31·23-s + 1.05·27-s − 0.813·29-s + 0.760·31-s − 0.281·33-s − 0.289·37-s + 0.579·39-s + 1.24·41-s − 1.29·43-s − 0.686·47-s − 0.945·49-s + 1.09·51-s + 0.529·53-s − 1.17·57-s + 0.490·59-s + 0.907·61-s + 0.0297·63-s − 1.23·69-s + 0.0346·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 1.61T + 3T^{2} \) |
| 7 | \( 1 + 0.618T + 7T^{2} \) |
| 13 | \( 1 + 2.23T + 13T^{2} \) |
| 17 | \( 1 + 4.85T + 17T^{2} \) |
| 19 | \( 1 - 5.47T + 19T^{2} \) |
| 23 | \( 1 - 6.32T + 23T^{2} \) |
| 29 | \( 1 + 4.38T + 29T^{2} \) |
| 31 | \( 1 - 4.23T + 31T^{2} \) |
| 37 | \( 1 + 1.76T + 37T^{2} \) |
| 41 | \( 1 - 7.94T + 41T^{2} \) |
| 43 | \( 1 + 8.47T + 43T^{2} \) |
| 47 | \( 1 + 4.70T + 47T^{2} \) |
| 53 | \( 1 - 3.85T + 53T^{2} \) |
| 59 | \( 1 - 3.76T + 59T^{2} \) |
| 61 | \( 1 - 7.09T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 0.291T + 71T^{2} \) |
| 73 | \( 1 - 7.09T + 73T^{2} \) |
| 79 | \( 1 + 2.85T + 79T^{2} \) |
| 83 | \( 1 + 1.14T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 + 4.90T + 97T^{2} \) |
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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
−7.972321652646191899294169880787, −6.90648630052925052043293278891, −6.72719423866405775219272896113, −5.67427604621533079136543898545, −5.14480233478941973416880134364, −4.42993826358224840032729771733, −3.32822225619184883394663681188, −2.49236762044894579170384434637, −1.15993521018487971398670594626, 0,
1.15993521018487971398670594626, 2.49236762044894579170384434637, 3.32822225619184883394663681188, 4.42993826358224840032729771733, 5.14480233478941973416880134364, 5.67427604621533079136543898545, 6.72719423866405775219272896113, 6.90648630052925052043293278891, 7.972321652646191899294169880787
