L(s) = 1 | + (−2.60 + 4.49i)3-s + (−8.45 + 14.6i)5-s + (−13.4 − 23.4i)9-s + (−45.7 + 26.4i)11-s − 49.8i·13-s + (−43.7 − 76.1i)15-s + (−0.687 − 1.19i)17-s + (62.9 + 36.3i)19-s + (−166. − 96.4i)23-s + (−80.4 − 139. i)25-s + (140. + 0.841i)27-s − 34.2i·29-s + (81.2 − 46.8i)31-s + (0.549 − 274. i)33-s + (−150. + 260. i)37-s + ⋯ |
L(s) = 1 | + (−0.501 + 0.865i)3-s + (−0.756 + 1.30i)5-s + (−0.496 − 0.868i)9-s + (−1.25 + 0.724i)11-s − 1.06i·13-s + (−0.753 − 1.31i)15-s + (−0.00980 − 0.0169i)17-s + (0.759 + 0.438i)19-s + (−1.51 − 0.874i)23-s + (−0.643 − 1.11i)25-s + (0.999 + 0.00599i)27-s − 0.219i·29-s + (0.470 − 0.271i)31-s + (0.00289 − 1.44i)33-s + (−0.667 + 1.15i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.445i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.4167766191\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4167766191\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.60 - 4.49i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (8.45 - 14.6i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (45.7 - 26.4i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 49.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (0.687 + 1.19i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-62.9 - 36.3i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (166. + 96.4i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 34.2iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-81.2 + 46.8i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (150. - 260. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 88.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 136.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (283. - 490. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-395. + 228. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-126. - 219. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-215. - 124. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-54.3 - 94.0i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 21.4iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-310. + 179. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-597. + 1.03e3i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 834.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (114. - 199. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.14e3iT - 9.12e5T^{2} \) |
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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
−10.25116970439169735178963630641, −9.876147417156753594652062254360, −8.237716025004515623991425778776, −7.64244234541269630414677027887, −6.55160747016438890153482607347, −5.59116343193401678053699490480, −4.55506708105514727251829781257, −3.46583852480576973123342091650, −2.63576121239269458569383000619, −0.18156428564882735174228685136,
0.814850654763183821912687837467, 2.12175220834669884314118067513, 3.76948539684692706291147090578, 5.03030314581641765836362199244, 5.57820238856737231425860751457, 6.86342570367146475353115060347, 7.84163786475809463183476453727, 8.336703738551843345350633565142, 9.286310730260612886956396644709, 10.53752538100237445577193975409