| L(s) = 1 | − i·3-s + 3i·7-s − 9-s − 2·11-s − i·13-s + 2i·17-s − 5·19-s + 3·21-s + 6i·23-s + i·27-s − 10·29-s + 3·31-s + 2i·33-s + 2i·37-s − 39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.13i·7-s − 0.333·9-s − 0.603·11-s − 0.277i·13-s + 0.485i·17-s − 1.14·19-s + 0.654·21-s + 1.25i·23-s + 0.192i·27-s − 1.85·29-s + 0.538·31-s + 0.348i·33-s + 0.328i·37-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7012699701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7012699701\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 - 3iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 17iT - 97T^{2} \) |
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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
−9.949907055488745137097779037514, −9.064836626535550633354138233983, −8.340335244204874851631296247914, −7.66888427676943697701842596601, −6.64656603942778753571332266857, −5.77248322643970656018602756557, −5.19405509753699174454130377229, −3.78438288556302106727629890536, −2.63293869948519873765064966244, −1.72634591722671636220940902706,
0.28079064034016865538553634585, 2.10770564300735768058513110529, 3.40900263759224670531452537491, 4.29625234040689046689858087905, 4.99944718464177753990442637794, 6.16619618781521749947669039417, 7.02800850792565103921338730538, 7.86234145614925000217120166161, 8.716295803944429446485180625998, 9.578495447981886821356180020888
