| L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + i·7-s + i·8-s + 2·9-s − 11-s − i·12-s + 2i·13-s + 14-s + 16-s + 3i·17-s − 2i·18-s + 19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.377i·7-s + 0.353i·8-s + 0.666·9-s − 0.301·11-s − 0.288i·12-s + 0.554i·13-s + 0.267·14-s + 0.250·16-s + 0.727i·17-s − 0.471i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33805 + 0.315872i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33805 + 0.315872i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - iT - 3T^{2} \) |
| 7 | \( 1 - iT - 7T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + 5iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−10.70052935863693205582412608707, −10.08249976273010348228620565460, −9.319854329964202764286319839755, −8.463749577853724317391471207347, −7.37042880314954080290418005632, −6.12329181017474829968448083531, −4.95841472221385591412573262785, −4.13532968303152766448208274082, −3.01684704977875269560812752016, −1.57408299339195360315909896913,
0.896359628440551867678730090357, 2.74837020184173054437806877058, 4.27887363484550370919956388633, 5.16987592127404206622876614398, 6.46125834300147038564548383913, 7.02366628534888717609421395882, 7.951039987261469923004506691235, 8.642158841752548457605784681454, 9.980048753757673560965407634009, 10.40484017622222170638886349028
