| L(s) = 1 | + (0.100 + 1.41i)2-s + (4.61 + 1.00i)3-s + (−1.97 + 0.284i)4-s + (1.70 + 4.69i)5-s + (−0.949 + 6.60i)6-s + (−4.20 + 7.70i)7-s + (−0.601 − 2.76i)8-s + (12.0 + 5.51i)9-s + (−6.45 + 2.88i)10-s + (−8.13 − 9.38i)11-s + (−9.41 − 0.673i)12-s + (−0.616 − 1.12i)13-s + (−11.2 − 5.15i)14-s + (3.16 + 23.3i)15-s + (3.83 − 1.12i)16-s + (−8.81 − 11.7i)17-s + ⋯ |
| L(s) = 1 | + (0.0504 + 0.705i)2-s + (1.53 + 0.334i)3-s + (−0.494 + 0.0711i)4-s + (0.341 + 0.939i)5-s + (−0.158 + 1.10i)6-s + (−0.600 + 1.10i)7-s + (−0.0751 − 0.345i)8-s + (1.34 + 0.612i)9-s + (−0.645 + 0.288i)10-s + (−0.739 − 0.853i)11-s + (−0.784 − 0.0561i)12-s + (−0.0474 − 0.0868i)13-s + (−0.806 − 0.368i)14-s + (0.211 + 1.55i)15-s + (0.239 − 0.0704i)16-s + (−0.518 − 0.692i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.453 - 0.891i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.24645 + 2.03223i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24645 + 2.03223i\) |
| \(L(2)\) |
|
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 | \( 1 + (-1.70 - 4.69i)T \) |
| 23 | \( 1 + (-22.9 + 1.02i)T \) |
| good | 3 | \( 1 + (-4.61 - 1.00i)T + (8.18 + 3.73i)T^{2} \) |
| 7 | \( 1 + (4.20 - 7.70i)T + (-26.4 - 41.2i)T^{2} \) |
| 11 | \( 1 + (8.13 + 9.38i)T + (-17.2 + 119. i)T^{2} \) |
| 13 | \( 1 + (0.616 + 1.12i)T + (-91.3 + 142. i)T^{2} \) |
| 17 | \( 1 + (8.81 + 11.7i)T + (-81.4 + 277. i)T^{2} \) |
| 19 | \( 1 + (-26.9 + 3.88i)T + (346. - 101. i)T^{2} \) |
| 29 | \( 1 + (-47.5 - 6.83i)T + (806. + 236. i)T^{2} \) |
| 31 | \( 1 + (-17.4 - 11.1i)T + (399. + 874. i)T^{2} \) |
| 37 | \( 1 + (18.2 + 48.8i)T + (-1.03e3 + 896. i)T^{2} \) |
| 41 | \( 1 + (-20.6 - 45.2i)T + (-1.10e3 + 1.27e3i)T^{2} \) |
| 43 | \( 1 + (11.5 + 2.50i)T + (1.68e3 + 768. i)T^{2} \) |
| 47 | \( 1 + (53.7 - 53.7i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-10.0 - 5.50i)T + (1.51e3 + 2.36e3i)T^{2} \) |
| 59 | \( 1 + (-9.51 + 32.3i)T + (-2.92e3 - 1.88e3i)T^{2} \) |
| 61 | \( 1 + (6.34 + 4.07i)T + (1.54e3 + 3.38e3i)T^{2} \) |
| 67 | \( 1 + (6.38 + 89.3i)T + (-4.44e3 + 638. i)T^{2} \) |
| 71 | \( 1 + (-56.6 + 65.3i)T + (-717. - 4.98e3i)T^{2} \) |
| 73 | \( 1 + (16.3 - 21.8i)T + (-1.50e3 - 5.11e3i)T^{2} \) |
| 79 | \( 1 + (23.0 - 78.4i)T + (-5.25e3 - 3.37e3i)T^{2} \) |
| 83 | \( 1 + (-39.5 + 14.7i)T + (5.20e3 - 4.51e3i)T^{2} \) |
| 89 | \( 1 + (51.8 + 80.7i)T + (-3.29e3 + 7.20e3i)T^{2} \) |
| 97 | \( 1 + (94.9 + 35.3i)T + (7.11e3 + 6.16e3i)T^{2} \) |
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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
−12.67007067520501754400351808016, −11.18179599186053438703103208277, −9.889639354154111265378984781576, −9.240452023183260332992874036898, −8.417989924153025962767677183103, −7.41792757409083173622845238221, −6.30760548583633443695529980422, −5.06872881071914128933813594272, −3.12121402235820007848546451996, −2.80150536224598357789124287572,
1.21084487057401842031132611407, 2.60479000221068142945003908603, 3.81127119152245701708038790236, 4.94474378541579959562471349199, 6.90436328938365514789872531108, 7.958646470453490453029013279756, 8.789049945374240699210776089413, 9.810309509061149787260598949339, 10.24881173787296776538562368439, 11.96506695521174970005748306469
