| L(s) = 1 | + i·2-s − 4-s + i·5-s + 5·7-s − i·8-s − 3·9-s − 10-s + 3·11-s + 2i·13-s + 5i·14-s + 16-s − i·17-s − 3i·18-s + 2i·19-s − i·20-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.447i·5-s + 1.88·7-s − 0.353i·8-s − 9-s − 0.316·10-s + 0.904·11-s + 0.554i·13-s + 1.33i·14-s + 0.250·16-s − 0.242i·17-s − 0.707i·18-s + 0.458i·19-s − 0.223i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12137 + 0.949949i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12137 + 0.949949i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 5 | \( 1 - iT \) |
| 37 | \( 1 + (-1 + 6i)T \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 7 | \( 1 - 5T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + iT - 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 5iT - 29T^{2} \) |
| 31 | \( 1 - iT - 31T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 + 11iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 + 7iT - 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 4iT - 89T^{2} \) |
| 97 | \( 1 + 11iT - 97T^{2} \) |
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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
−11.38535450603358434949651587528, −11.01813855821274131465918536013, −9.502988323329158951080709023435, −8.638622250309714165735668203535, −7.86666365125470232879567897046, −6.95509674041225330471271564641, −5.75614913427035035115577633899, −4.91001198159744368286971336078, −3.66799652141793241573627985898, −1.76595702036829419059033839363,
1.20415248561010885495150745839, 2.60439181039286588324359421768, 4.26241489100573263738454712543, 5.00510690401806604401580998990, 6.15982169094809552272739788908, 7.920384719287853773541256781167, 8.403454162043797059519129602577, 9.279632933407168197259545241829, 10.53704253329823385867232383739, 11.42731315511054187933035705937
