L(s) = 1 | + (7.54 − 13.0i)5-s + (−16.2 − 8.80i)7-s + (−8.56 + 4.94i)11-s + 67.8i·13-s + (−35.0 − 60.7i)17-s + (53.2 + 30.7i)19-s + (−113. − 65.7i)23-s + (−51.3 − 88.8i)25-s + 158. i·29-s + (66.2 − 38.2i)31-s + (−237. + 146. i)35-s + (−174. + 301. i)37-s − 138.·41-s − 539.·43-s + (111. − 193. i)47-s + ⋯ |
L(s) = 1 | + (0.674 − 1.16i)5-s + (−0.879 − 0.475i)7-s + (−0.234 + 0.135i)11-s + 1.44i·13-s + (−0.500 − 0.866i)17-s + (0.642 + 0.371i)19-s + (−1.03 − 0.596i)23-s + (−0.410 − 0.711i)25-s + 1.01i·29-s + (0.383 − 0.221i)31-s + (−1.14 + 0.707i)35-s + (−0.774 + 1.34i)37-s − 0.529·41-s − 1.91·43-s + (0.347 − 0.601i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00687 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.00687 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8023540000\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8023540000\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (16.2 + 8.80i)T \) |
good | 5 | \( 1 + (-7.54 + 13.0i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (8.56 - 4.94i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 67.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (35.0 + 60.7i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-53.2 - 30.7i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (113. + 65.7i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 158. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-66.2 + 38.2i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (174. - 301. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 138.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 539.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-111. + 193. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-459. + 265. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-271. - 470. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (116. + 67.0i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-160. - 277. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 416. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-472. + 272. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (161. - 279. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 885.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (812. - 1.40e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 739. iT - 9.12e5T^{2} \) |
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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
−9.808280329919996473742396511629, −9.009062090271730999320084419866, −8.382385956039919155583350344477, −7.04567049214586337566318331136, −6.52057618823607088743683533917, −5.35416336257269281896691081480, −4.64001289305476631691448006383, −3.61118741938061576141615408759, −2.21632304431702596510494852634, −1.13108315112134765924231403713,
0.20329511529711058526391770846, 2.05922466842020152335237969274, 2.94417010839944640435688490311, 3.68772700283238590542661202675, 5.35217685232125175023123169575, 6.00281038154797560933616023978, 6.67507314319607763042093813789, 7.65104468678088146509038263487, 8.545036696638233528076531317182, 9.634133798215477003905924248323