| L(s) = 1 | + 2-s + 4-s − 5-s − 4·7-s + 8-s − 10-s + 6·11-s + 13-s − 4·14-s + 16-s + 6·17-s + 2·19-s − 20-s + 6·22-s − 6·23-s + 25-s + 26-s − 4·28-s + 6·29-s + 2·31-s + 32-s + 6·34-s + 4·35-s + 2·37-s + 2·38-s − 40-s + 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s + 0.353·8-s − 0.316·10-s + 1.80·11-s + 0.277·13-s − 1.06·14-s + 1/4·16-s + 1.45·17-s + 0.458·19-s − 0.223·20-s + 1.27·22-s − 1.25·23-s + 1/5·25-s + 0.196·26-s − 0.755·28-s + 1.11·29-s + 0.359·31-s + 0.176·32-s + 1.02·34-s + 0.676·35-s + 0.328·37-s + 0.324·38-s − 0.158·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.309443794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.309443794\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.750626614872720057400968667768, −9.151877785951146734411403956564, −7.995215963426200914412120114699, −7.07291414966298812217331341983, −6.29345621975886340324442998465, −5.78605651391475461678969846796, −4.28250035270549478393778080408, −3.68007626878963657540665398656, −2.87374848181607439503651528979, −1.10216868918993511634847670128,
1.10216868918993511634847670128, 2.87374848181607439503651528979, 3.68007626878963657540665398656, 4.28250035270549478393778080408, 5.78605651391475461678969846796, 6.29345621975886340324442998465, 7.07291414966298812217331341983, 7.995215963426200914412120114699, 9.151877785951146734411403956564, 9.750626614872720057400968667768
