L(s) = 1 | + (−5.32 − 1.90i)2-s + 21.6·3-s + (24.7 + 20.2i)4-s + (36.8 − 42.0i)5-s + (−115. − 41.2i)6-s + 236. i·7-s + (−93.4 − 155. i)8-s + 227.·9-s + (−276. + 154. i)10-s − 192. i·11-s + (537. + 439. i)12-s + 975.·13-s + (449. − 1.25e3i)14-s + (798. − 911. i)15-s + (203. + 1.00e3i)16-s − 670. i·17-s + ⋯ |
L(s) = 1 | + (−0.941 − 0.336i)2-s + 1.39·3-s + (0.774 + 0.632i)4-s + (0.658 − 0.752i)5-s + (−1.31 − 0.467i)6-s + 1.82i·7-s + (−0.516 − 0.856i)8-s + 0.934·9-s + (−0.873 + 0.487i)10-s − 0.480i·11-s + (1.07 + 0.880i)12-s + 1.60·13-s + (0.612 − 1.71i)14-s + (0.916 − 1.04i)15-s + (0.198 + 0.980i)16-s − 0.563i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.82212 - 0.161435i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.82212 - 0.161435i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (5.32 + 1.90i)T \) |
| 5 | \( 1 + (-36.8 + 42.0i)T \) |
good | 3 | \( 1 - 21.6T + 243T^{2} \) |
| 7 | \( 1 - 236. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 192. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 975.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 670. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 456. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 2.02e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 2.78e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 963.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.72e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.81e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 254.T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.97e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 2.30e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.46e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 189. iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 2.49e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.82e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.47e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 6.97e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.33e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.56e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.81e4iT - 8.58e9T^{2} \) |
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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
−15.43629928640353066505646848465, −13.82805044971652061657228484482, −12.75597161999559697880517851960, −11.43141523677934018527328749707, −9.499807467673270578790623352376, −8.872530549199879935551826470766, −8.219353043520250821144791808673, −5.94866974512898288363901330257, −3.08464121552711284296559426505, −1.77250721948521934961081912149,
1.60562943543405900870669203094, 3.51752156988372951310452382058, 6.54925853698597363173505274151, 7.62005399253964578010403571182, 8.812579491747654387799728543676, 10.16749257544563221855679117237, 10.79540109516690711874732337597, 13.41517363650773640051406944693, 14.14742862346561858593989323935, 15.02139199389594434650377243382