| L(s) = 1 | + 3-s − 7-s − 2·9-s − 3·11-s + 6·13-s − 21-s − 2·23-s − 5·27-s − 6·29-s − 3·33-s + 37-s + 6·39-s − 9·41-s + 10·43-s − 47-s − 6·49-s − 53-s − 12·61-s + 2·63-s − 2·69-s − 5·71-s − 3·73-s + 3·77-s + 16·79-s + 81-s − 11·83-s − 6·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s − 2/3·9-s − 0.904·11-s + 1.66·13-s − 0.218·21-s − 0.417·23-s − 0.962·27-s − 1.11·29-s − 0.522·33-s + 0.164·37-s + 0.960·39-s − 1.40·41-s + 1.52·43-s − 0.145·47-s − 6/7·49-s − 0.137·53-s − 1.53·61-s + 0.251·63-s − 0.240·69-s − 0.593·71-s − 0.351·73-s + 0.341·77-s + 1.80·79-s + 1/9·81-s − 1.20·83-s − 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.114012092838855617855541915884, −7.70625654754739324427973981815, −6.59142287086646253480079955685, −5.91256197936574831748831189231, −5.28783982846590555348828148734, −4.07809925616778205203576578610, −3.38038987829739509149679811859, −2.65248198584538983022437228731, −1.56243220908714011720021399373, 0,
1.56243220908714011720021399373, 2.65248198584538983022437228731, 3.38038987829739509149679811859, 4.07809925616778205203576578610, 5.28783982846590555348828148734, 5.91256197936574831748831189231, 6.59142287086646253480079955685, 7.70625654754739324427973981815, 8.114012092838855617855541915884
