| L(s) = 1 | + 3-s − 7-s − 2·9-s + 2·11-s − 3·13-s − 21-s + 23-s − 5·25-s − 5·27-s + 29-s + 5·31-s + 2·33-s − 8·37-s − 3·39-s − 7·41-s + 4·43-s − 3·47-s + 49-s − 12·53-s − 4·59-s − 6·61-s + 2·63-s + 12·67-s + 69-s − 13·71-s + 3·73-s − 5·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.603·11-s − 0.832·13-s − 0.218·21-s + 0.208·23-s − 25-s − 0.962·27-s + 0.185·29-s + 0.898·31-s + 0.348·33-s − 1.31·37-s − 0.480·39-s − 1.09·41-s + 0.609·43-s − 0.437·47-s + 1/7·49-s − 1.64·53-s − 0.520·59-s − 0.768·61-s + 0.251·63-s + 1.46·67-s + 0.120·69-s − 1.54·71-s + 0.351·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.508210856755379642666895279911, −7.86703814559501307981524583380, −7.00186305873747291360095589864, −6.25546008812784414914556339354, −5.39754862952736690859916773578, −4.45716162893346661816411061782, −3.46697586319946797774089312924, −2.76949872597604657028021260692, −1.70693843272421829472577435864, 0,
1.70693843272421829472577435864, 2.76949872597604657028021260692, 3.46697586319946797774089312924, 4.45716162893346661816411061782, 5.39754862952736690859916773578, 6.25546008812784414914556339354, 7.00186305873747291360095589864, 7.86703814559501307981524583380, 8.508210856755379642666895279911
