L(s) = 1 | − 2.49·2-s − 3-s + 4.20·4-s − 1.71·5-s + 2.49·6-s + 7-s − 5.49·8-s + 9-s + 4.26·10-s − 3·11-s − 4.20·12-s + 1.20·13-s − 2.49·14-s + 1.71·15-s + 5.26·16-s + 2.49·17-s − 2.49·18-s − 19-s − 7.20·20-s − 21-s + 7.47·22-s + 7.47·23-s + 5.49·24-s − 2.06·25-s − 3·26-s − 27-s + 4.20·28-s + ⋯ |
L(s) = 1 | − 1.76·2-s − 0.577·3-s + 2.10·4-s − 0.766·5-s + 1.01·6-s + 0.377·7-s − 1.94·8-s + 0.333·9-s + 1.34·10-s − 0.904·11-s − 1.21·12-s + 0.334·13-s − 0.665·14-s + 0.442·15-s + 1.31·16-s + 0.604·17-s − 0.587·18-s − 0.229·19-s − 1.61·20-s − 0.218·21-s + 1.59·22-s + 1.55·23-s + 1.12·24-s − 0.412·25-s − 0.588·26-s − 0.192·27-s + 0.794·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 193 | \( 1 + T \) |
good | 2 | \( 1 + 2.49T + 2T^{2} \) |
| 5 | \( 1 + 1.71T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 1.20T + 13T^{2} \) |
| 17 | \( 1 - 2.49T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 7.47T + 23T^{2} \) |
| 29 | \( 1 + 2.22T + 29T^{2} \) |
| 31 | \( 1 + 0.0637T + 31T^{2} \) |
| 37 | \( 1 + 0.0637T + 37T^{2} \) |
| 41 | \( 1 + 5.41T + 41T^{2} \) |
| 43 | \( 1 + 0.936T + 43T^{2} \) |
| 47 | \( 1 - 2.64T + 47T^{2} \) |
| 53 | \( 1 + 6.77T + 53T^{2} \) |
| 59 | \( 1 + 9.26T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 + 4.47T + 67T^{2} \) |
| 71 | \( 1 + 9.18T + 71T^{2} \) |
| 73 | \( 1 + 4.61T + 73T^{2} \) |
| 79 | \( 1 + 0.140T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 - 7.20T + 89T^{2} \) |
| 97 | \( 1 + 3.54T + 97T^{2} \) |
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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
−10.34662799054845299615719069946, −9.381319435541491782358012072992, −8.464299276487119940130721612921, −7.73217450685809061484570551660, −7.14485305668778484838957450098, −5.98418736156878715155159806000, −4.75033806039549903062951810546, −3.09123110207663081906554914272, −1.48029708320423910213754158820, 0,
1.48029708320423910213754158820, 3.09123110207663081906554914272, 4.75033806039549903062951810546, 5.98418736156878715155159806000, 7.14485305668778484838957450098, 7.73217450685809061484570551660, 8.464299276487119940130721612921, 9.381319435541491782358012072992, 10.34662799054845299615719069946