L(s) = 1 | + 2·2-s + (−4.66 + 2.28i)3-s + 4·4-s + 8.40i·5-s + (−9.33 + 4.56i)6-s + 31.0·7-s + 8·8-s + (16.5 − 21.3i)9-s + 16.8i·10-s − 8.36·11-s + (−18.6 + 9.13i)12-s + 33.8i·13-s + 62.0·14-s + (−19.1 − 39.2i)15-s + 16·16-s − 52.8i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.898 + 0.439i)3-s + 0.5·4-s + 0.751i·5-s + (−0.635 + 0.310i)6-s + 1.67·7-s + 0.353·8-s + (0.614 − 0.789i)9-s + 0.531i·10-s − 0.229·11-s + (−0.449 + 0.219i)12-s + 0.722i·13-s + 1.18·14-s + (−0.330 − 0.674i)15-s + 0.250·16-s − 0.753i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.629771026\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.629771026\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + (4.66 - 2.28i)T \) |
| 59 | \( 1 + (-339. + 300. i)T \) |
good | 5 | \( 1 - 8.40iT - 125T^{2} \) |
| 7 | \( 1 - 31.0T + 343T^{2} \) |
| 11 | \( 1 + 8.36T + 1.33e3T^{2} \) |
| 13 | \( 1 - 33.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 52.8iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 13.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 124.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 146. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 205. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 278. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 402. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 466. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 513.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 100. iT - 1.48e5T^{2} \) |
| 61 | \( 1 - 413. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 771. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 262. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 355. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 942.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 109.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.05e3iT - 9.12e5T^{2} \) |
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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
−11.38325325703864704356808677137, −10.72217149734481024838109076607, −9.655530598871898207768311191997, −8.240858293388881210545823615359, −7.10056356546096217517759141641, −6.32098331631885328338326404461, −4.91157939286520356846898891349, −4.67910970733398316256703845457, −3.05481441726442668060125732148, −1.42705714054288958484571695931,
0.936440284693372449691342705523, 2.06255499721919867952571223837, 4.12890244021140087525796055185, 5.20248958263714282552753649857, 5.48428799017544503960922272710, 6.98836808245197000564253519604, 7.88859276195312802476481017661, 8.740685941846896154055052822318, 10.50809549658316493596936833783, 11.00488421289326895321693554094