L(s) = 1 | + (2.60 − 4.50i)2-s + (−9.52 − 16.4i)4-s + (−2.5 − 4.33i)5-s + (−12.2 + 21.1i)7-s − 57.4·8-s − 26.0·10-s + (−14.4 + 25.1i)11-s + (32.6 + 56.6i)13-s + (63.4 + 109. i)14-s + (−73.2 + 126. i)16-s − 68.1·17-s + 104.·19-s + (−47.6 + 82.4i)20-s + (75.3 + 130. i)22-s + (−77.4 − 134. i)23-s + ⋯ |
L(s) = 1 | + (0.919 − 1.59i)2-s + (−1.19 − 2.06i)4-s + (−0.223 − 0.387i)5-s + (−0.658 + 1.14i)7-s − 2.53·8-s − 0.822·10-s + (−0.397 + 0.688i)11-s + (0.697 + 1.20i)13-s + (1.21 + 2.09i)14-s + (−1.14 + 1.98i)16-s − 0.972·17-s + 1.26·19-s + (−0.532 + 0.922i)20-s + (0.730 + 1.26i)22-s + (−0.701 − 1.21i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8756072511\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8756072511\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.5 + 4.33i)T \) |
good | 2 | \( 1 + (-2.60 + 4.50i)T + (-4 - 6.92i)T^{2} \) |
| 7 | \( 1 + (12.2 - 21.1i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (14.4 - 25.1i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-32.6 - 56.6i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 68.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 104.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (77.4 + 134. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (102. - 178. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-9.12 - 15.8i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 337.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (97.9 + 169. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (167. - 290. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (2.50 - 4.33i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 319.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-215. - 372. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (297. - 514. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (97.9 + 169. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 425.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 929.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (12.2 - 21.1i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (272. - 472. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 84.1T + 7.04e5T^{2} \) |
| 97 | \( 1 + (413. - 716. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−11.14391754753700368717054264396, −10.18817761758887606000441427938, −9.251448068262493473307825734451, −8.725506296378688709741378879385, −6.80355076160951773215539440948, −5.64688845676059111291399017992, −4.76057891308172857997710867637, −3.76166591245201679644262890966, −2.60821841377420355361848665918, −1.62474088093926047367559464018,
0.20917988851622874198767539088, 3.34377090671405162206838438883, 3.79127065343419629751175203851, 5.22898878675605127457248327251, 6.03398827116268117935322505160, 6.97073589207881097377120362914, 7.67526332343942880095191301375, 8.416562591837205285775580450276, 9.756944080094272382340522223566, 10.81844426164035141891002322403