| L(s) = 1 | − 2-s − 4-s − 5-s + 2·7-s + 3·8-s + 10-s − 4·11-s − 13-s − 2·14-s − 16-s − 4·17-s + 6·19-s + 20-s + 4·22-s + 25-s + 26-s − 2·28-s − 4·29-s − 10·31-s − 5·32-s + 4·34-s − 2·35-s − 2·37-s − 6·38-s − 3·40-s − 6·41-s − 8·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 0.755·7-s + 1.06·8-s + 0.316·10-s − 1.20·11-s − 0.277·13-s − 0.534·14-s − 1/4·16-s − 0.970·17-s + 1.37·19-s + 0.223·20-s + 0.852·22-s + 1/5·25-s + 0.196·26-s − 0.377·28-s − 0.742·29-s − 1.79·31-s − 0.883·32-s + 0.685·34-s − 0.338·35-s − 0.328·37-s − 0.973·38-s − 0.474·40-s − 0.937·41-s − 1.21·43-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 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−10.19964286858850027386838092871, −9.332189727210250988718420123449, −8.455924495253961688482406234394, −7.77509555057385964870974093324, −7.09926588277152631380243196233, −5.33227627105416183888148278982, −4.80291034122253544985154871503, −3.48108877573719190338767515003, −1.80936597723698845777463745934, 0,
1.80936597723698845777463745934, 3.48108877573719190338767515003, 4.80291034122253544985154871503, 5.33227627105416183888148278982, 7.09926588277152631380243196233, 7.77509555057385964870974093324, 8.455924495253961688482406234394, 9.332189727210250988718420123449, 10.19964286858850027386838092871
