| L(s) = 1 | + (−0.624 − 1.92i)2-s + (1.69 − 1.23i)3-s + (3.16 − 2.29i)4-s + (−9.66 − 5.61i)5-s + (−3.43 − 2.49i)6-s + 24.6·7-s + (−19.4 − 14.1i)8-s + (−6.98 + 21.4i)9-s + (−4.76 + 22.1i)10-s + (21.6 + 66.5i)11-s + (2.53 − 7.80i)12-s + (4.46 − 13.7i)13-s + (−15.4 − 47.4i)14-s + (−23.3 + 2.38i)15-s + (−5.38 + 16.5i)16-s + (9.49 + 6.89i)17-s + ⋯ |
| L(s) = 1 | + (−0.220 − 0.680i)2-s + (0.326 − 0.237i)3-s + (0.395 − 0.287i)4-s + (−0.864 − 0.502i)5-s + (−0.233 − 0.169i)6-s + 1.33·7-s + (−0.861 − 0.625i)8-s + (−0.258 + 0.795i)9-s + (−0.150 + 0.699i)10-s + (0.592 + 1.82i)11-s + (0.0610 − 0.187i)12-s + (0.0953 − 0.293i)13-s + (−0.294 − 0.906i)14-s + (−0.401 + 0.0411i)15-s + (−0.0842 + 0.259i)16-s + (0.135 + 0.0983i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.940286 - 0.717909i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.940286 - 0.717909i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (9.66 + 5.61i)T \) |
| good | 2 | \( 1 + (0.624 + 1.92i)T + (-6.47 + 4.70i)T^{2} \) |
| 3 | \( 1 + (-1.69 + 1.23i)T + (8.34 - 25.6i)T^{2} \) |
| 7 | \( 1 - 24.6T + 343T^{2} \) |
| 11 | \( 1 + (-21.6 - 66.5i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (-4.46 + 13.7i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-9.49 - 6.89i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (59.4 + 43.1i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (38.8 + 119. i)T + (-9.84e3 + 7.15e3i)T^{2} \) |
| 29 | \( 1 + (49.7 - 36.1i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-18.6 - 13.5i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (42.8 - 131. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-105. + 324. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 302.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-65.5 + 47.6i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (29.7 - 21.6i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (57.6 - 177. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-131. - 403. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (712. + 517. i)T + (9.29e4 + 2.86e5i)T^{2} \) |
| 71 | \( 1 + (-316. + 230. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-141. - 434. i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (101. - 73.7i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (181. + 131. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + (156. + 482. i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-1.11e3 + 810. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−16.99748052353318630281121487399, −15.35580948884159121889990184702, −14.57666917810571116975193760381, −12.61150043640944547825980539293, −11.64246374799509091917330084790, −10.49239227361674831212866391613, −8.662174619630350319370285402751, −7.31838168798817079932942517080, −4.66971254448262010413227486864, −1.89140240036913638218276524520,
3.55284685792622481587993141937, 6.19932846795064329126703773639, 7.88105424340173673366608306161, 8.710468243092135542668816756339, 11.22578542897574869722298198706, 11.75510685775146751763678745954, 14.23670010738346289569136579310, 14.92138167087089579757456517053, 16.05394741449420428402595672399, 17.17426197564546218915849261327
