| L(s) = 1 | − 2-s + 4-s + 2·5-s − 2·7-s − 8-s − 2·10-s − 11-s − 13-s + 2·14-s + 16-s − 4·17-s + 2·19-s + 2·20-s + 22-s + 4·23-s − 25-s + 26-s − 2·28-s + 4·29-s − 6·31-s − 32-s + 4·34-s − 4·35-s − 2·37-s − 2·38-s − 2·40-s − 2·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.755·7-s − 0.353·8-s − 0.632·10-s − 0.301·11-s − 0.277·13-s + 0.534·14-s + 1/4·16-s − 0.970·17-s + 0.458·19-s + 0.447·20-s + 0.213·22-s + 0.834·23-s − 1/5·25-s + 0.196·26-s − 0.377·28-s + 0.742·29-s − 1.07·31-s − 0.176·32-s + 0.685·34-s − 0.676·35-s − 0.328·37-s − 0.324·38-s − 0.316·40-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.808698849675691766212827075476, −7.71932727251282984481455338783, −6.98926476421663655514923634446, −6.31442723092302762375922021584, −5.58544482064785584334065954203, −4.65033596988990663125202314865, −3.32882189373931439230976853189, −2.52043965207579637103123721935, −1.52679182118429669252511440021, 0,
1.52679182118429669252511440021, 2.52043965207579637103123721935, 3.32882189373931439230976853189, 4.65033596988990663125202314865, 5.58544482064785584334065954203, 6.31442723092302762375922021584, 6.98926476421663655514923634446, 7.71932727251282984481455338783, 8.808698849675691766212827075476
