| L(s) = 1 | + (−1.28 + 0.587i)2-s + (−0.810 − 2.75i)3-s + (1.30 − 1.51i)4-s + (4.95 − 0.644i)5-s + (2.66 + 3.07i)6-s + (−0.976 − 6.79i)7-s + (−0.796 + 2.71i)8-s + (0.612 − 0.393i)9-s + (−5.99 + 3.74i)10-s + (15.1 + 6.90i)11-s + (−5.23 − 2.38i)12-s + (−16.4 − 2.36i)13-s + (5.24 + 8.16i)14-s + (−5.79 − 13.1i)15-s + (−0.569 − 3.95i)16-s + (5.15 + 5.95i)17-s + ⋯ |
| L(s) = 1 | + (−0.643 + 0.293i)2-s + (−0.270 − 0.919i)3-s + (0.327 − 0.377i)4-s + (0.991 − 0.128i)5-s + (0.443 + 0.512i)6-s + (−0.139 − 0.970i)7-s + (−0.0996 + 0.339i)8-s + (0.0680 − 0.0437i)9-s + (−0.599 + 0.374i)10-s + (1.37 + 0.627i)11-s + (−0.436 − 0.199i)12-s + (−1.26 − 0.181i)13-s + (0.374 + 0.583i)14-s + (−0.386 − 0.877i)15-s + (−0.0355 − 0.247i)16-s + (0.303 + 0.350i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00848 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.00848 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.855985 - 0.848750i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.855985 - 0.848750i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.28 - 0.587i)T \) |
| 5 | \( 1 + (-4.95 + 0.644i)T \) |
| 23 | \( 1 + (-9.37 + 21.0i)T \) |
| good | 3 | \( 1 + (0.810 + 2.75i)T + (-7.57 + 4.86i)T^{2} \) |
| 7 | \( 1 + (0.976 + 6.79i)T + (-47.0 + 13.8i)T^{2} \) |
| 11 | \( 1 + (-15.1 - 6.90i)T + (79.2 + 91.4i)T^{2} \) |
| 13 | \( 1 + (16.4 + 2.36i)T + (162. + 47.6i)T^{2} \) |
| 17 | \( 1 + (-5.15 - 5.95i)T + (-41.1 + 286. i)T^{2} \) |
| 19 | \( 1 + (8.26 + 7.16i)T + (51.3 + 357. i)T^{2} \) |
| 29 | \( 1 + (29.7 + 34.3i)T + (-119. + 832. i)T^{2} \) |
| 31 | \( 1 + (-23.0 - 6.75i)T + (808. + 519. i)T^{2} \) |
| 37 | \( 1 + (22.6 - 14.5i)T + (568. - 1.24e3i)T^{2} \) |
| 41 | \( 1 + (58.9 + 37.8i)T + (698. + 1.52e3i)T^{2} \) |
| 43 | \( 1 + (-73.8 + 21.6i)T + (1.55e3 - 9.99e2i)T^{2} \) |
| 47 | \( 1 - 13.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-10.5 - 73.3i)T + (-2.69e3 + 791. i)T^{2} \) |
| 59 | \( 1 + (-7.25 + 50.4i)T + (-3.33e3 - 980. i)T^{2} \) |
| 61 | \( 1 + (32.2 - 109. i)T + (-3.13e3 - 2.01e3i)T^{2} \) |
| 67 | \( 1 + (-12.4 - 27.3i)T + (-2.93e3 + 3.39e3i)T^{2} \) |
| 71 | \( 1 + (-25.3 - 55.4i)T + (-3.30e3 + 3.80e3i)T^{2} \) |
| 73 | \( 1 + (42.4 + 36.7i)T + (758. + 5.27e3i)T^{2} \) |
| 79 | \( 1 + (85.9 + 12.3i)T + (5.98e3 + 1.75e3i)T^{2} \) |
| 83 | \( 1 + (-6.98 + 4.49i)T + (2.86e3 - 6.26e3i)T^{2} \) |
| 89 | \( 1 + (-25.6 - 87.3i)T + (-6.66e3 + 4.28e3i)T^{2} \) |
| 97 | \( 1 + (-115. - 74.4i)T + (3.90e3 + 8.55e3i)T^{2} \) |
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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
−11.90831096678541338288156766470, −10.45809556255065815246818745477, −9.824462075820033004288665635360, −8.878299916254070774828017353875, −7.36571384069747829255807686124, −6.89629504318384238981343202622, −5.97602747777757421249801292176, −4.39434618922380527520292151986, −2.10773677026330452717681302639, −0.873983647748596398834087313785,
1.83296086597003228451242616507, 3.34778238955284630286009828163, 4.96576461216873717414237997805, 5.99489703224906045422853135415, 7.18647323951670900345837091994, 8.846165118120610407220360996247, 9.438673727352421490028090145980, 10.03979765175657316376991510411, 11.11976498641190630022371467991, 11.94958691427424227073252603264
