| L(s) = 1 | + (1.28 + 0.587i)2-s + (−1.62 + 5.54i)3-s + (1.30 + 1.51i)4-s + (0.750 + 4.94i)5-s + (−5.35 + 6.17i)6-s + (−0.543 + 3.77i)7-s + (0.796 + 2.71i)8-s + (−20.5 − 13.1i)9-s + (−1.93 + 6.80i)10-s + (15.1 − 6.92i)11-s + (−10.5 + 4.80i)12-s + (6.09 − 0.876i)13-s + (−2.91 + 4.54i)14-s + (−28.6 − 3.88i)15-s + (−0.569 + 3.95i)16-s + (8.15 − 9.40i)17-s + ⋯ |
| L(s) = 1 | + (0.643 + 0.293i)2-s + (−0.542 + 1.84i)3-s + (0.327 + 0.377i)4-s + (0.150 + 0.988i)5-s + (−0.892 + 1.02i)6-s + (−0.0775 + 0.539i)7-s + (0.0996 + 0.339i)8-s + (−2.28 − 1.46i)9-s + (−0.193 + 0.680i)10-s + (1.37 − 0.629i)11-s + (−0.876 + 0.400i)12-s + (0.468 − 0.0674i)13-s + (−0.208 + 0.324i)14-s + (−1.90 − 0.259i)15-s + (−0.0355 + 0.247i)16-s + (0.479 − 0.553i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.104i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.994 - 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0994140 + 1.90011i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0994140 + 1.90011i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.28 - 0.587i)T \) |
| 5 | \( 1 + (-0.750 - 4.94i)T \) |
| 23 | \( 1 + (-1.00 - 22.9i)T \) |
| good | 3 | \( 1 + (1.62 - 5.54i)T + (-7.57 - 4.86i)T^{2} \) |
| 7 | \( 1 + (0.543 - 3.77i)T + (-47.0 - 13.8i)T^{2} \) |
| 11 | \( 1 + (-15.1 + 6.92i)T + (79.2 - 91.4i)T^{2} \) |
| 13 | \( 1 + (-6.09 + 0.876i)T + (162. - 47.6i)T^{2} \) |
| 17 | \( 1 + (-8.15 + 9.40i)T + (-41.1 - 286. i)T^{2} \) |
| 19 | \( 1 + (-18.5 + 16.1i)T + (51.3 - 357. i)T^{2} \) |
| 29 | \( 1 + (26.1 - 30.1i)T + (-119. - 832. i)T^{2} \) |
| 31 | \( 1 + (37.5 - 11.0i)T + (808. - 519. i)T^{2} \) |
| 37 | \( 1 + (17.7 + 11.4i)T + (568. + 1.24e3i)T^{2} \) |
| 41 | \( 1 + (-29.7 + 19.0i)T + (698. - 1.52e3i)T^{2} \) |
| 43 | \( 1 + (3.53 + 1.03i)T + (1.55e3 + 9.99e2i)T^{2} \) |
| 47 | \( 1 + 20.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (2.70 - 18.7i)T + (-2.69e3 - 791. i)T^{2} \) |
| 59 | \( 1 + (7.82 + 54.4i)T + (-3.33e3 + 980. i)T^{2} \) |
| 61 | \( 1 + (-27.3 - 93.2i)T + (-3.13e3 + 2.01e3i)T^{2} \) |
| 67 | \( 1 + (-30.8 + 67.4i)T + (-2.93e3 - 3.39e3i)T^{2} \) |
| 71 | \( 1 + (-15.0 + 33.0i)T + (-3.30e3 - 3.80e3i)T^{2} \) |
| 73 | \( 1 + (22.4 - 19.4i)T + (758. - 5.27e3i)T^{2} \) |
| 79 | \( 1 + (-83.0 + 11.9i)T + (5.98e3 - 1.75e3i)T^{2} \) |
| 83 | \( 1 + (12.2 + 7.84i)T + (2.86e3 + 6.26e3i)T^{2} \) |
| 89 | \( 1 + (-31.9 + 108. i)T + (-6.66e3 - 4.28e3i)T^{2} \) |
| 97 | \( 1 + (142. - 91.6i)T + (3.90e3 - 8.55e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.02706643009763440691105383623, −11.32629707096471399832662013320, −10.80400404822701549567775297434, −9.475950735815975675130335116863, −8.996769252442583912850069693749, −7.08246674107666282316441217672, −5.88212405733844908033039587229, −5.30511663539481518759914218422, −3.76217967682705799662649361519, −3.22551789406868649001961192708,
0.986563200830738821651045690561, 1.85453924298846892218452780004, 3.96786026389108908449032423999, 5.49665906838221728192338440838, 6.28775045237999868919457721294, 7.28142570777955758522560945442, 8.249775211623786748727415657487, 9.579782292886093034751531083160, 11.07174600165806388320136088325, 11.96694557081278941940173367515
