| L(s) = 1 | − 2.41i·2-s + 2i·3-s − 3.82·4-s + 4.82·6-s + 0.828i·7-s + 4.41i·8-s − 9-s − 4.82·11-s − 7.65i·12-s + 2i·13-s + 1.99·14-s + 2.99·16-s − 2.82i·17-s + 2.41i·18-s − 0.828·19-s + ⋯ |
| L(s) = 1 | − 1.70i·2-s + 1.15i·3-s − 1.91·4-s + 1.97·6-s + 0.313i·7-s + 1.56i·8-s − 0.333·9-s − 1.45·11-s − 2.21i·12-s + 0.554i·13-s + 0.534·14-s + 0.749·16-s − 0.685i·17-s + 0.569i·18-s − 0.190·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.522416 + 0.322871i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.522416 + 0.322871i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.41iT - 2T^{2} \) |
| 3 | \( 1 - 2iT - 3T^{2} \) |
| 7 | \( 1 - 0.828iT - 7T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 2.82iT - 17T^{2} \) |
| 19 | \( 1 + 0.828T + 19T^{2} \) |
| 23 | \( 1 - 8.82iT - 23T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 - 8.48iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + 0.343iT - 47T^{2} \) |
| 53 | \( 1 + 7.65iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 7.65T + 61T^{2} \) |
| 67 | \( 1 + 10.4iT - 67T^{2} \) |
| 71 | \( 1 - 7.31T + 71T^{2} \) |
| 73 | \( 1 - 8.48iT - 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 + 12.8iT - 83T^{2} \) |
| 89 | \( 1 + 3.65T + 89T^{2} \) |
| 97 | \( 1 - 4.48iT - 97T^{2} \) |
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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
−10.58398900159999843837438598884, −9.791916847456475860649503817538, −9.429539126246900236565855659121, −8.477990042359555698814321120745, −7.25187404212121187614104464163, −5.41054621581820714585244244112, −4.87357471908976139441332310904, −3.78146777540943074370420425455, −3.03205754067048600552143009051, −1.82655675637076912049312911936,
0.30620756861729516670729676068, 2.30385375557248273607590224548, 4.09903684341191952450336855298, 5.31323269119523513982193402981, 5.95619437309432876618056402143, 6.98877572090141888045934799083, 7.41880192803429992872452724615, 8.197256247927807152361095799086, 8.778722661615424846564226863038, 10.17569668213723855639152987294
