L(s) = 1 | + (3.30 − 5.73i)5-s + (4.31 + 18.0i)7-s + (1.54 − 0.893i)11-s − 4.72i·13-s + (−23.8 − 41.3i)17-s + (40.1 + 23.1i)19-s + (30.2 + 17.4i)23-s + (40.5 + 70.3i)25-s − 48.3i·29-s + (107. − 62.2i)31-s + (117. + 34.8i)35-s + (−137. + 238. i)37-s − 37.3·41-s + 215.·43-s + (−53.1 + 91.9i)47-s + ⋯ |
L(s) = 1 | + (0.295 − 0.512i)5-s + (0.233 + 0.972i)7-s + (0.0424 − 0.0244i)11-s − 0.100i·13-s + (−0.340 − 0.589i)17-s + (0.484 + 0.279i)19-s + (0.273 + 0.158i)23-s + (0.324 + 0.562i)25-s − 0.309i·29-s + (0.624 − 0.360i)31-s + (0.567 + 0.168i)35-s + (−0.612 + 1.06i)37-s − 0.142·41-s + 0.765·43-s + (−0.164 + 0.285i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.165031406\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.165031406\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (-4.31 - 18.0i)T \) |
good | 5 | \( 1 + (-3.30 + 5.73i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-1.54 + 0.893i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 4.72iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (23.8 + 41.3i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-40.1 - 23.1i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-30.2 - 17.4i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 48.3iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-107. + 62.2i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (137. - 238. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 37.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 215.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (53.1 - 91.9i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-233. + 134. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-149. - 259. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (292. + 168. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-188. - 326. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 816. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (596. - 344. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-307. + 531. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.32e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-480. + 832. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 449. iT - 9.12e5T^{2} \) |
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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
−9.508104386720309702373525896056, −8.920030485228128879616189100273, −8.164585459465417571934872010295, −7.18562729627760183842341496849, −6.12356984870777461274567850786, −5.33487524963745899512060831374, −4.62036722244861305235213530230, −3.23718844161184543437415443562, −2.20136515423552836861003727037, −1.01032954475689477198995200077,
0.64047097418829949895897269665, 1.92463111002306514720998933163, 3.14565720952116872067096815711, 4.14795139093372696610794062092, 5.06760654087212339222307745150, 6.24607481395541027859195721036, 6.94285040724381762764203626287, 7.70834653790136932811239562327, 8.671406727828233705281150529657, 9.553142734711825449128751799450