L(s) = 1 | + 1.73i·3-s + (0.5 − 2.59i)7-s − 2.99·9-s + 1.73i·13-s − 5.19i·19-s + (4.5 + 0.866i)21-s − 5.19i·27-s − 1.73i·31-s + 10·37-s − 2.99·39-s + 13·43-s + (−6.5 − 2.59i)49-s + 9·57-s − 15.5i·61-s + (−1.49 + 7.79i)63-s + ⋯ |
L(s) = 1 | + 0.999i·3-s + (0.188 − 0.981i)7-s − 0.999·9-s + 0.480i·13-s − 1.19i·19-s + (0.981 + 0.188i)21-s − 0.999i·27-s − 0.311i·31-s + 1.64·37-s − 0.480·39-s + 1.98·43-s + (−0.928 − 0.371i)49-s + 1.19·57-s − 1.99i·61-s + (−0.188 + 0.981i)63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.579785580\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.579785580\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 2.59i)T \) |
good | 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 1.73iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 5.19iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 1.73iT - 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 13T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 15.5iT - 61T^{2} \) |
| 67 | \( 1 - 11T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 13.8iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 5.19iT - 97T^{2} \) |
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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
−9.392370219469317633615496673953, −8.357552991546846011215925798387, −7.60400890942188797442488002613, −6.71876485610259097606633427180, −5.85173560484428553557316107271, −4.77508339988735229769785415241, −4.31715088607608757116860871668, −3.41818524781695930131604683720, −2.34302864968474590573016515114, −0.65572117082124692658368154963,
1.08698661313618801790473441789, 2.23658816712640536133639029121, 2.97881154299357734548419582509, 4.23080757060767761464277536192, 5.58437134721207395232452429461, 5.80401584568793894548997609694, 6.79019970009611985461886590635, 7.71648291966824864121746920651, 8.212791670058767593650587445511, 8.959996768408671496646051101715