L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.5 − 0.866i)3-s + (0.766 + 0.642i)4-s + (−0.766 + 0.642i)6-s + (−0.500 − 0.866i)8-s + (−0.499 − 0.866i)9-s + (−1.70 − 0.984i)11-s + (0.939 − 0.342i)12-s + (0.173 + 0.984i)16-s + (0.173 + 0.984i)18-s + (−0.766 − 0.642i)19-s + (1.26 + 1.50i)22-s − 24-s + (−0.766 − 0.642i)25-s − 0.999·27-s + ⋯ |
L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.5 − 0.866i)3-s + (0.766 + 0.642i)4-s + (−0.766 + 0.642i)6-s + (−0.500 − 0.866i)8-s + (−0.499 − 0.866i)9-s + (−1.70 − 0.984i)11-s + (0.939 − 0.342i)12-s + (0.173 + 0.984i)16-s + (0.173 + 0.984i)18-s + (−0.766 − 0.642i)19-s + (1.26 + 1.50i)22-s − 24-s + (−0.766 − 0.642i)25-s − 0.999·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5861709345\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5861709345\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.766 + 0.642i)T \) |
good | 5 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + (1.70 + 0.984i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 23 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 29 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \) |
| 43 | \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 53 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 59 | \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \) |
| 61 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 67 | \( 1 + (0.233 + 0.642i)T + (-0.766 + 0.642i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 79 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + 1.96iT - T^{2} \) |
| 89 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 97 | \( 1 + (0.439 - 1.20i)T + (-0.766 - 0.642i)T^{2} \) |
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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
−9.150726187032358779225512205557, −8.679661450188846427457921439528, −7.83436041887298095709913020198, −7.44124281122044666527785703402, −6.35350497601198216286438898822, −5.60040544689009230958976090728, −3.93346284435676047901186448853, −2.76330371898249011864684337846, −2.24173585129427490355437791862, −0.57457456778215855691634205879,
2.03699543536134656839832326714, 2.83338289210601886688805026828, 4.25071783665429212909391063875, 5.21046010727415924409143605851, 5.94667955171198608780899742732, 7.24204171147477328841470679506, 7.916841557327709696445423456565, 8.405226869307436373210219853199, 9.603639352674730421743013692188, 9.773603757201584289164578102375