| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 6·11-s − 4·13-s + 14-s + 16-s + 19-s − 6·22-s − 6·23-s − 5·25-s − 4·26-s + 28-s − 6·29-s − 4·31-s + 32-s + 2·37-s + 38-s + 6·41-s + 8·43-s − 6·44-s − 6·46-s − 12·47-s + 49-s − 5·50-s − 4·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 1.80·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.229·19-s − 1.27·22-s − 1.25·23-s − 25-s − 0.784·26-s + 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.328·37-s + 0.162·38-s + 0.937·41-s + 1.21·43-s − 0.904·44-s − 0.884·46-s − 1.75·47-s + 1/7·49-s − 0.707·50-s − 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−8.253388728553679418104012227756, −7.72267429824042640830912324171, −7.19656696119957322087841996812, −5.92878595009151954682954682188, −5.41336725514925878422215774510, −4.67735326515217802644123677936, −3.76220617340977541587938712566, −2.62977888879285736786115861970, −1.98341656088136381985063969955, 0,
1.98341656088136381985063969955, 2.62977888879285736786115861970, 3.76220617340977541587938712566, 4.67735326515217802644123677936, 5.41336725514925878422215774510, 5.92878595009151954682954682188, 7.19656696119957322087841996812, 7.72267429824042640830912324171, 8.253388728553679418104012227756
