L(s) = 1 | + 2·2-s − 3-s + 2·4-s + 3·5-s − 2·6-s − 7-s + 9-s + 6·10-s + 6·11-s − 2·12-s − 2·13-s − 2·14-s − 3·15-s − 4·16-s + 5·17-s + 2·18-s − 4·19-s + 6·20-s + 21-s + 12·22-s + 7·23-s + 4·25-s − 4·26-s − 27-s − 2·28-s − 4·29-s − 6·30-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s + 1.34·5-s − 0.816·6-s − 0.377·7-s + 1/3·9-s + 1.89·10-s + 1.80·11-s − 0.577·12-s − 0.554·13-s − 0.534·14-s − 0.774·15-s − 16-s + 1.21·17-s + 0.471·18-s − 0.917·19-s + 1.34·20-s + 0.218·21-s + 2.55·22-s + 1.45·23-s + 4/5·25-s − 0.784·26-s − 0.192·27-s − 0.377·28-s − 0.742·29-s − 1.09·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 753 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 753 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.265861381\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.265861381\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 251 | \( 1 - T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−10.49283560494082341031718378462, −9.477625738331160526696068083368, −9.035524211302788217619155515720, −7.16523629506177851040872285197, −6.38886902485816053562142678104, −5.87504705361469674255388266396, −5.03537659855798816622686416266, −4.05910781778142341786989265122, −2.95420573130777934179957267826, −1.55286639694222212264713507650,
1.55286639694222212264713507650, 2.95420573130777934179957267826, 4.05910781778142341786989265122, 5.03537659855798816622686416266, 5.87504705361469674255388266396, 6.38886902485816053562142678104, 7.16523629506177851040872285197, 9.035524211302788217619155515720, 9.477625738331160526696068083368, 10.49283560494082341031718378462