L(s) = 1 | − 2-s + 4-s − 5-s − 3.69i·7-s − 8-s + 10-s − 5.10·11-s − 6.05i·13-s + 3.69i·14-s + 16-s + 0.252i·17-s + 4.17·19-s − 20-s + 5.10·22-s − 5.65i·23-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s − 1.39i·7-s − 0.353·8-s + 0.316·10-s − 1.53·11-s − 1.67i·13-s + 0.987i·14-s + 0.250·16-s + 0.0613i·17-s + 0.958·19-s − 0.223·20-s + 1.08·22-s − 1.17i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.216i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6838050854\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6838050854\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (-6.06 + 5.50i)T \) |
good | 7 | \( 1 + 3.69iT - 7T^{2} \) |
| 11 | \( 1 + 5.10T + 11T^{2} \) |
| 13 | \( 1 + 6.05iT - 13T^{2} \) |
| 17 | \( 1 - 0.252iT - 17T^{2} \) |
| 19 | \( 1 - 4.17T + 19T^{2} \) |
| 23 | \( 1 + 5.65iT - 23T^{2} \) |
| 29 | \( 1 - 9.12iT - 29T^{2} \) |
| 31 | \( 1 + 7.22iT - 31T^{2} \) |
| 37 | \( 1 + 5.83T + 37T^{2} \) |
| 41 | \( 1 - 9.99T + 41T^{2} \) |
| 43 | \( 1 + 5.80iT - 43T^{2} \) |
| 47 | \( 1 + 13.0iT - 47T^{2} \) |
| 53 | \( 1 - 3.38T + 53T^{2} \) |
| 59 | \( 1 + 11.2iT - 59T^{2} \) |
| 61 | \( 1 - 2.77iT - 61T^{2} \) |
| 71 | \( 1 + 8.07iT - 71T^{2} \) |
| 73 | \( 1 - 2.89T + 73T^{2} \) |
| 79 | \( 1 - 8.22iT - 79T^{2} \) |
| 83 | \( 1 + 12.6iT - 83T^{2} \) |
| 89 | \( 1 - 2.69iT - 89T^{2} \) |
| 97 | \( 1 + 3.99iT - 97T^{2} \) |
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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
−7.57419329204520476360822344682, −7.43884456381455319600463753669, −6.51656837183856859614836741461, −5.41631060330330561746240231873, −5.00115691847268467169852833789, −3.79134987463981221836576492488, −3.18998897354217928025226972941, −2.28819133574807858305806706141, −0.817409462981344406168754422746, −0.30037344327477239605526274229,
1.35795928755614698052524343930, 2.41388490246006319885683707192, 2.87893050398894210369011199549, 4.05200077824769249002244323124, 5.02631211449817440375362884721, 5.64535244084765695992380060097, 6.35710126188640928405229606980, 7.31962195348394612855817165623, 7.75688656371042491833869105669, 8.476738466082890843744085347849