| L(s) = 1 | + i·2-s − 4-s + (−2 − i)5-s − 4i·7-s − i·8-s + (1 − 2i)10-s − 2·11-s + i·13-s + 4·14-s + 16-s + 4i·17-s − 2·19-s + (2 + i)20-s − 2i·22-s + 6i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.894 − 0.447i)5-s − 1.51i·7-s − 0.353i·8-s + (0.316 − 0.632i)10-s − 0.603·11-s + 0.277i·13-s + 1.06·14-s + 0.250·16-s + 0.970i·17-s − 0.458·19-s + (0.447 + 0.223i)20-s − 0.426i·22-s + 1.25i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4013569602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4013569602\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 14T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 18T + 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−10.06821688937809707127692894756, −9.194637400753628967105753285744, −8.151017020020790484714756705318, −7.70159059975857772822177930474, −7.02830232270319661159796847170, −6.02489362238179160017318955529, −4.87520326704765102938360227493, −4.15677630159668975394585451273, −3.41919559060861455219490890985, −1.31108925300982847587427098913,
0.18476061369210921516229013215, 2.30146465391094410906662143949, 2.86928905207467419687942198982, 4.02866380992653245545035593663, 5.07101407121827046552598680767, 5.86880338955461021212004951152, 7.05738593769258433706371900339, 7.964217839984763862300972731444, 8.748423491934475628755849833790, 9.330818633964789614253912923314
