| L(s) = 1 | + 2-s − 4-s + 5-s − 4·7-s − 3·8-s + 10-s − 2·11-s − 13-s − 4·14-s − 16-s − 2·17-s − 6·19-s − 20-s − 2·22-s + 6·23-s + 25-s − 26-s + 4·28-s − 2·29-s − 10·31-s + 5·32-s − 2·34-s − 4·35-s − 2·37-s − 6·38-s − 3·40-s + 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s − 1.51·7-s − 1.06·8-s + 0.316·10-s − 0.603·11-s − 0.277·13-s − 1.06·14-s − 1/4·16-s − 0.485·17-s − 1.37·19-s − 0.223·20-s − 0.426·22-s + 1.25·23-s + 1/5·25-s − 0.196·26-s + 0.755·28-s − 0.371·29-s − 1.79·31-s + 0.883·32-s − 0.342·34-s − 0.676·35-s − 0.328·37-s − 0.973·38-s − 0.474·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−10.24076128195437772789084342069, −9.238371497422229278947323218386, −8.903866275348653436076253219750, −7.36945570129601409328488904448, −6.35831289645710659676663930685, −5.66957165960173077695586779200, −4.59731852050444182415870036677, −3.51994456797006266949922249859, −2.53409816778618263405222233425, 0,
2.53409816778618263405222233425, 3.51994456797006266949922249859, 4.59731852050444182415870036677, 5.66957165960173077695586779200, 6.35831289645710659676663930685, 7.36945570129601409328488904448, 8.903866275348653436076253219750, 9.238371497422229278947323218386, 10.24076128195437772789084342069
