L(s) = 1 | + i·2-s + 3-s − 4-s + 3i·5-s + i·6-s − i·7-s − i·8-s − 2·9-s − 3·10-s + 6i·11-s − 12-s + 14-s + 3i·15-s + 16-s + 3·17-s − 2i·18-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s + 1.34i·5-s + 0.408i·6-s − 0.377i·7-s − 0.353i·8-s − 0.666·9-s − 0.948·10-s + 1.80i·11-s − 0.288·12-s + 0.267·14-s + 0.774i·15-s + 0.250·16-s + 0.727·17-s − 0.471i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.643949 + 1.20323i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.643949 + 1.20323i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 13 | \( 1 \) |
good | 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 - 3iT - 5T^{2} \) |
| 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 - 6iT - 11T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 14iT - 67T^{2} \) |
| 71 | \( 1 - 3iT - 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−11.90001938655246642161124448681, −10.68951228124473413516396354642, −9.970806440026238400885315201692, −9.026687183871298662204841434953, −7.77756593576225514711222864746, −7.19424624485425059378327960563, −6.33863028403466947556436283928, −4.94658345547980238986834259477, −3.61364056707757015726153296503, −2.42549519787529325700206170165,
0.955561772345536587695390546105, 2.72840214598476615671868185953, 3.80091000690223610783049956115, 5.21954265905149702204053665503, 5.96957496143121267328805667713, 8.085411398851038431044936968833, 8.524661109040951760923924459211, 9.161260634330154200706631741996, 10.25682981267986509509626086362, 11.47962023623496683249556426007