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 | \( 1 + (142. + 91.6i)T + (3.90e3 + 8.55e3i)T^{2} \) |
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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
−11.96694557081278941940173367515, −11.07174600165806388320136088325, −9.579782292886093034751531083160, −8.249775211623786748727415657487, −7.28142570777955758522560945442, −6.28775045237999868919457721294, −5.49665906838221728192338440838, −3.96786026389108908449032423999, −1.85453924298846892218452780004, −0.986563200830738821651045690561,
3.22551789406868649001961192708, 3.76217967682705799662649361519, 5.30511663539481518759914218422, 5.88212405733844908033039587229, 7.08246674107666282316441217672, 8.996769252442583912850069693749, 9.475950735815975675130335116863, 10.80400404822701549567775297434, 11.32629707096471399832662013320, 12.02706643009763440691105383623