| L(s) = 1 | − 3-s − 2·5-s + 7-s + 9-s − 6·11-s + 2·15-s + 8·17-s − 4·19-s − 21-s + 2·23-s − 25-s − 27-s + 10·29-s + 7·31-s + 6·33-s − 2·35-s + 2·37-s − 4·41-s − 7·43-s − 2·45-s + 6·47-s − 6·49-s − 8·51-s + 12·55-s + 4·57-s − 6·59-s − 5·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.80·11-s + 0.516·15-s + 1.94·17-s − 0.917·19-s − 0.218·21-s + 0.417·23-s − 1/5·25-s − 0.192·27-s + 1.85·29-s + 1.25·31-s + 1.04·33-s − 0.338·35-s + 0.328·37-s − 0.624·41-s − 1.06·43-s − 0.298·45-s + 0.875·47-s − 6/7·49-s − 1.12·51-s + 1.61·55-s + 0.529·57-s − 0.781·59-s − 0.640·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−8.091700985821681590178555050111, −7.52571837192161150176551631284, −6.60704368067383593833331018664, −5.75279036253856303509931750246, −4.99054908977038406703092271541, −4.50016485827570000070247134735, −3.36179058572120127392854998904, −2.61334668592456949054868237213, −1.17372789775893167269749168742, 0,
1.17372789775893167269749168742, 2.61334668592456949054868237213, 3.36179058572120127392854998904, 4.50016485827570000070247134735, 4.99054908977038406703092271541, 5.75279036253856303509931750246, 6.60704368067383593833331018664, 7.52571837192161150176551631284, 8.091700985821681590178555050111
