| L(s) = 1 | + (−0.540 − 0.841i)2-s + (0.665 + 0.0957i)3-s + (−0.415 + 0.909i)4-s + (−0.171 − 2.22i)5-s + (−0.279 − 0.612i)6-s + (1.62 − 1.40i)7-s + (0.989 − 0.142i)8-s + (−2.44 − 0.717i)9-s + (−1.78 + 1.34i)10-s + (3.00 + 1.93i)11-s + (−0.363 + 0.566i)12-s + (−3.40 − 2.94i)13-s + (−2.06 − 0.605i)14-s + (0.0993 − 1.50i)15-s + (−0.654 − 0.755i)16-s + (5.12 − 2.34i)17-s + ⋯ |
| L(s) = 1 | + (−0.382 − 0.594i)2-s + (0.384 + 0.0552i)3-s + (−0.207 + 0.454i)4-s + (−0.0766 − 0.997i)5-s + (−0.114 − 0.249i)6-s + (0.613 − 0.531i)7-s + (0.349 − 0.0503i)8-s + (−0.814 − 0.239i)9-s + (−0.563 + 0.426i)10-s + (0.907 + 0.583i)11-s + (−0.105 + 0.163i)12-s + (−0.943 − 0.817i)13-s + (−0.550 − 0.161i)14-s + (0.0256 − 0.387i)15-s + (−0.163 − 0.188i)16-s + (1.24 − 0.567i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0541 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0541 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.762548 - 0.805029i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.762548 - 0.805029i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.540 + 0.841i)T \) |
| 5 | \( 1 + (0.171 + 2.22i)T \) |
| 23 | \( 1 + (2.94 - 3.78i)T \) |
| good | 3 | \( 1 + (-0.665 - 0.0957i)T + (2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (-1.62 + 1.40i)T + (0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-3.00 - 1.93i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (3.40 + 2.94i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-5.12 + 2.34i)T + (11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.28 + 5.00i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-3.52 - 7.71i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (0.681 + 4.74i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (2.34 - 7.99i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-9.99 + 2.93i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-3.35 - 0.481i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 10.7iT - 47T^{2} \) |
| 53 | \( 1 + (4.00 - 3.46i)T + (7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-1.25 + 1.45i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.973 - 6.77i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (0.464 + 0.723i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-7.34 + 4.72i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-5.78 - 2.64i)T + (47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (9.61 - 11.0i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-1.86 + 6.35i)T + (-69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (0.622 - 4.32i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (1.51 + 5.16i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−11.98376332964037456487845994057, −11.12184626552482468229401962405, −9.716491123395090117433522173264, −9.263403161099076783611733309543, −8.091589434251736622246472555119, −7.39111215873767102976619267536, −5.43728210947487738151362002700, −4.38636825466679576857629088810, −2.95856490290603479988580631299, −1.11292299360116677704305334679,
2.20800868978405972867055828281, 3.77989219493178442352667526195, 5.53209331155583144735600083735, 6.39602348063779267011342685813, 7.67256525811810941818891926137, 8.322858908543336009769078658697, 9.424836184541228726245216233728, 10.38508285488213818426829549122, 11.53489432516641033376905121931, 12.16132438071287555408849742859
