| L(s) = 1 | + i·5-s + 2·7-s − 2i·11-s + 2i·13-s − 4·17-s − 4i·19-s − 4·23-s − 25-s − 2i·29-s + 4·31-s + 2i·35-s + 2i·37-s − 6·41-s − 4i·43-s − 8·47-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s + 0.755·7-s − 0.603i·11-s + 0.554i·13-s − 0.970·17-s − 0.917i·19-s − 0.834·23-s − 0.200·25-s − 0.371i·29-s + 0.718·31-s + 0.338i·35-s + 0.328i·37-s − 0.937·41-s − 0.609i·43-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6441297310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6441297310\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−8.082400256523640018282581260327, −6.93142814549780539705180739273, −6.62110122072285913163900606985, −5.72619023313681491768728941686, −4.85240049707536555008275018235, −4.26960590191886573809103148557, −3.32090664175627645551676191427, −2.40616837131909052644371411417, −1.59956901693486321499320967536, −0.15633567195328539711829573712,
1.33931723470777923243463564057, 2.05179052624208101002207894400, 3.12591327207140560339997469979, 4.19103009978474118572067126364, 4.67909559982567454125530140772, 5.45883953832016902731987089983, 6.20615840572147053438454479915, 7.00292078738893923101586874100, 7.898336721004458663399858952717, 8.214237187798307473687710618935
