| L(s) = 1 | + (−2 + 3.46i)5-s + (−1.5 − 2.59i)7-s + (−14 − 24.2i)11-s + (5.5 − 9.52i)13-s + 44·17-s + 29·19-s + (−86 + 148. i)23-s + (54.5 + 94.3i)25-s + (−96 − 166. i)29-s + (−58 + 100. i)31-s + 12·35-s − 69·37-s + (−192 + 332. i)41-s + (−164 − 284. i)43-s + (−78 − 135. i)47-s + ⋯ |
| L(s) = 1 | + (−0.178 + 0.309i)5-s + (−0.0809 − 0.140i)7-s + (−0.383 − 0.664i)11-s + (0.117 − 0.203i)13-s + 0.627·17-s + 0.350·19-s + (−0.779 + 1.35i)23-s + (0.435 + 0.755i)25-s + (−0.614 − 1.06i)29-s + (−0.336 + 0.582i)31-s + 0.0579·35-s − 0.306·37-s + (−0.731 + 1.26i)41-s + (−0.581 − 1.00i)43-s + (−0.242 − 0.419i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3379888937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3379888937\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (2 - 3.46i)T + (-62.5 - 108. i)T^{2} \) |
| 7 | \( 1 + (1.5 + 2.59i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (14 + 24.2i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-5.5 + 9.52i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 44T + 4.91e3T^{2} \) |
| 19 | \( 1 - 29T + 6.85e3T^{2} \) |
| 23 | \( 1 + (86 - 148. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (96 + 166. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (58 - 100. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 69T + 5.06e4T^{2} \) |
| 41 | \( 1 + (192 - 332. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (164 + 284. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (78 + 135. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 392T + 1.48e5T^{2} \) |
| 59 | \( 1 + (206 - 356. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-212.5 - 368. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (128.5 - 222. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 1.00e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 359T + 3.89e5T^{2} \) |
| 79 | \( 1 + (438.5 + 759. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-164 - 284. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 1.57e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-741.5 - 1.28e3i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−10.47362736808719738266358984860, −9.803586214842984923001502768036, −8.766220573377818803061991285253, −7.83710455216326123862722090300, −7.15038691480305695229992723516, −5.94739116337177163680974560944, −5.22674783140479214483519109481, −3.78612797171542236881154582129, −3.02922855041636864156497544124, −1.46505242564859219650657255533,
0.095178750804725917289445888137, 1.67001074949305629272307819952, 2.95882826096766963076725284060, 4.22163461386539498193456493056, 5.09960772482325228198984868192, 6.15767905482969988938885064332, 7.16769827192836269803740190009, 8.051510484432067680178575118528, 8.870697933595930938982494850057, 9.820218308157106674603338489164
