| L(s) = 1 | + (1.28 − 0.587i)2-s + (−0.725 − 2.47i)3-s + (1.30 − 1.51i)4-s + (4.93 + 0.829i)5-s + (−2.38 − 2.75i)6-s + (−0.102 − 0.712i)7-s + (0.796 − 2.71i)8-s + (1.99 − 1.28i)9-s + (6.83 − 1.83i)10-s + (10.3 + 4.72i)11-s + (−4.68 − 2.13i)12-s + (−4.37 − 0.628i)13-s + (−0.550 − 0.856i)14-s + (−1.52 − 12.7i)15-s + (−0.569 − 3.95i)16-s + (−21.2 − 24.4i)17-s + ⋯ |
| L(s) = 1 | + (0.643 − 0.293i)2-s + (−0.241 − 0.823i)3-s + (0.327 − 0.377i)4-s + (0.986 + 0.165i)5-s + (−0.397 − 0.458i)6-s + (−0.0146 − 0.101i)7-s + (0.0996 − 0.339i)8-s + (0.221 − 0.142i)9-s + (0.683 − 0.183i)10-s + (0.940 + 0.429i)11-s + (−0.390 − 0.178i)12-s + (−0.336 − 0.0483i)13-s + (−0.0393 − 0.0611i)14-s + (−0.101 − 0.852i)15-s + (−0.0355 − 0.247i)16-s + (−1.24 − 1.43i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.166 + 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.166 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.88547 - 1.59308i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.88547 - 1.59308i\) |
| \(L(2)\) |
|
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 | \( 1 + (-4.93 - 0.829i)T \) |
| 23 | \( 1 + (-0.814 - 22.9i)T \) |
| good | 3 | \( 1 + (0.725 + 2.47i)T + (-7.57 + 4.86i)T^{2} \) |
| 7 | \( 1 + (0.102 + 0.712i)T + (-47.0 + 13.8i)T^{2} \) |
| 11 | \( 1 + (-10.3 - 4.72i)T + (79.2 + 91.4i)T^{2} \) |
| 13 | \( 1 + (4.37 + 0.628i)T + (162. + 47.6i)T^{2} \) |
| 17 | \( 1 + (21.2 + 24.4i)T + (-41.1 + 286. i)T^{2} \) |
| 19 | \( 1 + (3.60 + 3.11i)T + (51.3 + 357. i)T^{2} \) |
| 29 | \( 1 + (-24.5 - 28.3i)T + (-119. + 832. i)T^{2} \) |
| 31 | \( 1 + (32.2 + 9.45i)T + (808. + 519. i)T^{2} \) |
| 37 | \( 1 + (-19.5 + 12.5i)T + (568. - 1.24e3i)T^{2} \) |
| 41 | \( 1 + (-51.2 - 32.9i)T + (698. + 1.52e3i)T^{2} \) |
| 43 | \( 1 + (57.0 - 16.7i)T + (1.55e3 - 9.99e2i)T^{2} \) |
| 47 | \( 1 + 11.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-9.85 - 68.5i)T + (-2.69e3 + 791. i)T^{2} \) |
| 59 | \( 1 + (-2.06 + 14.3i)T + (-3.33e3 - 980. i)T^{2} \) |
| 61 | \( 1 + (17.8 - 60.8i)T + (-3.13e3 - 2.01e3i)T^{2} \) |
| 67 | \( 1 + (-41.1 - 90.0i)T + (-2.93e3 + 3.39e3i)T^{2} \) |
| 71 | \( 1 + (-12.0 - 26.3i)T + (-3.30e3 + 3.80e3i)T^{2} \) |
| 73 | \( 1 + (-54.0 - 46.8i)T + (758. + 5.27e3i)T^{2} \) |
| 79 | \( 1 + (106. + 15.3i)T + (5.98e3 + 1.75e3i)T^{2} \) |
| 83 | \( 1 + (13.3 - 8.57i)T + (2.86e3 - 6.26e3i)T^{2} \) |
| 89 | \( 1 + (14.4 + 49.2i)T + (-6.66e3 + 4.28e3i)T^{2} \) |
| 97 | \( 1 + (16.1 + 10.3i)T + (3.90e3 + 8.55e3i)T^{2} \) |
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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
−11.89288266160713879868617222891, −11.09755487776251526683021591499, −9.812048624683109491523017528642, −9.127762060294694185386804276150, −7.16887443705161413821794372112, −6.74666552544565452863117886973, −5.59643740068721383156971652741, −4.34119914509042976410418723497, −2.57930352524310943979073222182, −1.32989673421059857648302855375,
2.07009074602727609014773124951, 3.90869945504081296471837592110, 4.79096528434021060254014219736, 5.97605340744709310165363855493, 6.70913070293110368056748538748, 8.402803696760849860315697476106, 9.330980055893345273968425713915, 10.38315790037401147136589063860, 11.09347051006276489237157023434, 12.40072937892043879114425968439
