| L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s − 2i·7-s − i·8-s − 9-s − 4·11-s + i·12-s + 4i·13-s + 2·14-s + 16-s − i·17-s − i·18-s + 4·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.755i·7-s − 0.353i·8-s − 0.333·9-s − 1.20·11-s + 0.288i·12-s + 1.10i·13-s + 0.534·14-s + 0.250·16-s − 0.242i·17-s − 0.235i·18-s + 0.917·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.441732910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.441732910\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 + iT \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.819731164713835960317903770403, −7.892292402500814383671431939756, −7.37192899681605490894887814631, −6.93338551956464417723385448032, −5.84467766369760306801212006407, −5.25632789793723211254706713620, −4.27594077610886542899878596084, −3.34172748604609181180386642868, −2.10908779075671772190505074111, −0.797064277997278402066762760625,
0.70867531533723461617089106513, 2.46396108908156555758593884457, 2.84373588692901210370802614156, 3.92772055593496786690940044664, 4.98115030578720377870554750709, 5.42251122589078439659760444592, 6.29344026794985491995552079206, 7.62953912248559245935096759250, 8.262509490808275606086979037705, 8.883734070865607093545011730055
