| L(s) = 1 | + (−0.100 − 1.41i)2-s + (−4.28 − 0.933i)3-s + (−1.97 + 0.284i)4-s + (−1.41 + 4.79i)5-s + (−0.883 + 6.14i)6-s + (0.967 − 1.77i)7-s + (0.601 + 2.76i)8-s + (9.33 + 4.26i)9-s + (6.90 + 1.50i)10-s + (2.33 + 2.69i)11-s + (8.75 + 0.626i)12-s + (−0.750 − 1.37i)13-s + (−2.59 − 1.18i)14-s + (10.5 − 19.2i)15-s + (3.83 − 1.12i)16-s + (−8.69 − 11.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.0504 − 0.705i)2-s + (−1.42 − 0.311i)3-s + (−0.494 + 0.0711i)4-s + (−0.282 + 0.959i)5-s + (−0.147 + 1.02i)6-s + (0.138 − 0.253i)7-s + (0.0751 + 0.345i)8-s + (1.03 + 0.473i)9-s + (0.690 + 0.150i)10-s + (0.212 + 0.245i)11-s + (0.729 + 0.0521i)12-s + (−0.0577 − 0.105i)13-s + (−0.185 − 0.0846i)14-s + (0.702 − 1.28i)15-s + (0.239 − 0.0704i)16-s + (−0.511 − 0.683i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.712891 - 0.363016i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.712891 - 0.363016i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.100 + 1.41i)T \) |
| 5 | \( 1 + (1.41 - 4.79i)T \) |
| 23 | \( 1 + (-15.3 + 17.0i)T \) |
| good | 3 | \( 1 + (4.28 + 0.933i)T + (8.18 + 3.73i)T^{2} \) |
| 7 | \( 1 + (-0.967 + 1.77i)T + (-26.4 - 41.2i)T^{2} \) |
| 11 | \( 1 + (-2.33 - 2.69i)T + (-17.2 + 119. i)T^{2} \) |
| 13 | \( 1 + (0.750 + 1.37i)T + (-91.3 + 142. i)T^{2} \) |
| 17 | \( 1 + (8.69 + 11.6i)T + (-81.4 + 277. i)T^{2} \) |
| 19 | \( 1 + (-31.3 + 4.51i)T + (346. - 101. i)T^{2} \) |
| 29 | \( 1 + (0.538 + 0.0774i)T + (806. + 236. i)T^{2} \) |
| 31 | \( 1 + (-28.5 - 18.3i)T + (399. + 874. i)T^{2} \) |
| 37 | \( 1 + (13.3 + 35.7i)T + (-1.03e3 + 896. i)T^{2} \) |
| 41 | \( 1 + (-5.05 - 11.0i)T + (-1.10e3 + 1.27e3i)T^{2} \) |
| 43 | \( 1 + (-56.4 - 12.2i)T + (1.68e3 + 768. i)T^{2} \) |
| 47 | \( 1 + (-1.16 + 1.16i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-85.7 - 46.8i)T + (1.51e3 + 2.36e3i)T^{2} \) |
| 59 | \( 1 + (7.55 - 25.7i)T + (-2.92e3 - 1.88e3i)T^{2} \) |
| 61 | \( 1 + (-33.7 - 21.6i)T + (1.54e3 + 3.38e3i)T^{2} \) |
| 67 | \( 1 + (3.30 + 46.2i)T + (-4.44e3 + 638. i)T^{2} \) |
| 71 | \( 1 + (14.2 - 16.4i)T + (-717. - 4.98e3i)T^{2} \) |
| 73 | \( 1 + (-35.7 + 47.8i)T + (-1.50e3 - 5.11e3i)T^{2} \) |
| 79 | \( 1 + (-27.9 + 95.3i)T + (-5.25e3 - 3.37e3i)T^{2} \) |
| 83 | \( 1 + (58.3 - 21.7i)T + (5.20e3 - 4.51e3i)T^{2} \) |
| 89 | \( 1 + (-20.4 - 31.8i)T + (-3.29e3 + 7.20e3i)T^{2} \) |
| 97 | \( 1 + (32.8 + 12.2i)T + (7.11e3 + 6.16e3i)T^{2} \) |
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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
−11.75812431520767081601186674367, −10.96837981935975334931720666444, −10.40293878522631035042426255752, −9.210959829150920279891257932439, −7.51766412576868253245881553373, −6.78985541552210051406709626341, −5.56212724837430892018743760539, −4.41217266022151839255142163400, −2.81630798211556799945382142497, −0.809812665147341113623338434754,
0.909972935920332300166060111394, 4.02047756290889645549841706757, 5.16689088996673993115196077522, 5.68795106364787960696033141760, 6.89831260163731040921087597146, 8.125647913284419082007449502364, 9.169778241660190818661380645947, 10.14522507591383863742367343030, 11.43748628172573807197798503969, 11.91594144010264188904517914007
