L(s) = 1 | + (−2.5 + 4.33i)5-s + (11 + 19.0i)7-s + (−4.5 − 7.79i)11-s + (−8.5 + 14.7i)13-s + 75·17-s − 4·19-s + (91.5 − 158. i)23-s + (−12.5 − 21.6i)25-s + (64.5 + 111. i)29-s + (93.5 − 161. i)31-s − 110·35-s − 34·37-s + (132 − 228. i)41-s + (−221.5 − 383. i)43-s + (304.5 + 527. i)47-s + ⋯ |
L(s) = 1 | + (−0.223 + 0.387i)5-s + (0.593 + 1.02i)7-s + (−0.123 − 0.213i)11-s + (−0.181 + 0.314i)13-s + 1.07·17-s − 0.0482·19-s + (0.829 − 1.43i)23-s + (−0.100 − 0.173i)25-s + (0.413 + 0.715i)29-s + (0.541 − 0.938i)31-s − 0.531·35-s − 0.151·37-s + (0.502 − 0.870i)41-s + (−0.785 − 1.36i)43-s + (0.945 + 1.63i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.357105999\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.357105999\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.5 - 4.33i)T \) |
good | 7 | \( 1 + (-11 - 19.0i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (4.5 + 7.79i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (8.5 - 14.7i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 75T + 4.91e3T^{2} \) |
| 19 | \( 1 + 4T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-91.5 + 158. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-64.5 - 111. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-93.5 + 161. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 34T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-132 + 228. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (221.5 + 383. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-304.5 - 527. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 228T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-30 + 51.9i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-227 - 393. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-122 + 211. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 444T + 3.57e5T^{2} \) |
| 73 | \( 1 - 398T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-174.5 - 302. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-519 - 898. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 852T + 7.04e5T^{2} \) |
| 97 | \( 1 + (457 + 791. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−8.927160289944172325203777224848, −8.434849524621675052822974007102, −7.56352253279900984347434398898, −6.73405443485970310721049845661, −5.78772693843080610431410256309, −5.09518968601489752763870683793, −4.10719458588245643611938530978, −2.93671301088842772093354486414, −2.20335936984373662627277247895, −0.824035797658204898708101768928,
0.73221605902118199363277345128, 1.51156191971170039694481270940, 3.00673134517374779776956232836, 3.91717419039542342930871540096, 4.83151475442300871000364450268, 5.46817162028381580440138851295, 6.66622860099694755241341941637, 7.56378067924706852336035093852, 7.935457598451804144186668615434, 8.887021347989308177739519677577