| L(s) = 1 | + (0.0763 + 0.482i)2-s + (−1.01 + 0.517i)3-s + (1.67 − 0.544i)4-s + (1.47 − 1.67i)5-s + (−0.327 − 0.450i)6-s + (1.32 − 2.59i)7-s + (0.833 + 1.63i)8-s + (−1.00 + 1.37i)9-s + (0.922 + 0.583i)10-s + (−1.41 + 1.41i)12-s + (2.89 − 0.457i)13-s + (1.35 + 0.440i)14-s + (−0.630 + 2.46i)15-s + (2.12 − 1.54i)16-s + (−5.20 − 0.824i)17-s + (−0.740 − 0.377i)18-s + ⋯ |
| L(s) = 1 | + (0.0540 + 0.341i)2-s + (−0.586 + 0.298i)3-s + (0.837 − 0.272i)4-s + (0.660 − 0.750i)5-s + (−0.133 − 0.183i)6-s + (0.500 − 0.982i)7-s + (0.294 + 0.578i)8-s + (−0.333 + 0.458i)9-s + (0.291 + 0.184i)10-s + (−0.409 + 0.409i)12-s + (0.801 − 0.126i)13-s + (0.362 + 0.117i)14-s + (−0.162 + 0.637i)15-s + (0.531 − 0.385i)16-s + (−1.26 − 0.199i)17-s + (−0.174 − 0.0889i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.303i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 + 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.75203 - 0.272495i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.75203 - 0.272495i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.47 + 1.67i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (-0.0763 - 0.482i)T + (-1.90 + 0.618i)T^{2} \) |
| 3 | \( 1 + (1.01 - 0.517i)T + (1.76 - 2.42i)T^{2} \) |
| 7 | \( 1 + (-1.32 + 2.59i)T + (-4.11 - 5.66i)T^{2} \) |
| 13 | \( 1 + (-2.89 + 0.457i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (5.20 + 0.824i)T + (16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-1.26 + 3.89i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (2.12 + 2.12i)T + 23iT^{2} \) |
| 29 | \( 1 + (-0.817 - 2.51i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-5.45 - 3.96i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.03 - 0.528i)T + (21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (-3.38 - 1.09i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (5.07 - 5.07i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.67 - 3.28i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (0.231 + 1.45i)T + (-50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (-1.52 + 0.496i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-4.07 - 5.61i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.31 + 1.31i)T - 67iT^{2} \) |
| 71 | \( 1 + (2.32 - 1.68i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (13.1 + 6.71i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-12.3 - 8.96i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-1.04 + 6.60i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 - 11.1iT - 89T^{2} \) |
| 97 | \( 1 + (9.94 - 1.57i)T + (92.2 - 29.9i)T^{2} \) |
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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
−10.79009452150330922629797440425, −9.985954933880865453395650160477, −8.742405231162800793965336863733, −7.955354916870122663478013864372, −6.79797657491901690659144028445, −6.09908792607943596659370158066, −5.07601877931629940166085558894, −4.45899935070283901536930823934, −2.52669572303530907080987680935, −1.14602720564166103557908238891,
1.69761844082980232301542933814, 2.62070836582790414907711473203, 3.82822191067121584218833488087, 5.58971142147326514574824827615, 6.17847363493809140746854609607, 6.82135383862936776007625047804, 8.015925046918919712610061620400, 8.990111367279010438961556662312, 10.07857953334292299979817055329, 10.94346460851961244486534008183
