| L(s) = 1 | + (1.45 − 4.48i)3-s + (3.01 − 2.19i)5-s + (−2.51 + 0.817i)7-s + (−10.7 − 7.79i)9-s + (−5.49 − 9.52i)11-s + (4.83 − 6.64i)13-s + (−5.43 − 16.7i)15-s + (−1.42 − 1.96i)17-s + (22.1 + 7.20i)19-s + 12.4i·21-s − 28.1·23-s + (−3.43 + 10.5i)25-s + (−16.2 + 11.8i)27-s + (6.71 − 2.18i)29-s + (−17.9 − 13.0i)31-s + ⋯ |
| L(s) = 1 | + (0.486 − 1.49i)3-s + (0.603 − 0.438i)5-s + (−0.359 + 0.116i)7-s + (−1.19 − 0.866i)9-s + (−0.499 − 0.866i)11-s + (0.371 − 0.511i)13-s + (−0.362 − 1.11i)15-s + (−0.0838 − 0.115i)17-s + (1.16 + 0.379i)19-s + 0.594i·21-s − 1.22·23-s + (−0.137 + 0.422i)25-s + (−0.602 + 0.437i)27-s + (0.231 − 0.0752i)29-s + (−0.578 − 0.420i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.686635 - 1.69897i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.686635 - 1.69897i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.51 - 0.817i)T \) |
| 11 | \( 1 + (5.49 + 9.52i)T \) |
| good | 3 | \( 1 + (-1.45 + 4.48i)T + (-7.28 - 5.29i)T^{2} \) |
| 5 | \( 1 + (-3.01 + 2.19i)T + (7.72 - 23.7i)T^{2} \) |
| 13 | \( 1 + (-4.83 + 6.64i)T + (-52.2 - 160. i)T^{2} \) |
| 17 | \( 1 + (1.42 + 1.96i)T + (-89.3 + 274. i)T^{2} \) |
| 19 | \( 1 + (-22.1 - 7.20i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + 28.1T + 529T^{2} \) |
| 29 | \( 1 + (-6.71 + 2.18i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (17.9 + 13.0i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-3.08 - 9.49i)T + (-1.10e3 + 804. i)T^{2} \) |
| 41 | \( 1 + (-31.0 - 10.0i)T + (1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + 71.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (13.4 - 41.3i)T + (-1.78e3 - 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-55.6 - 40.4i)T + (868. + 2.67e3i)T^{2} \) |
| 59 | \( 1 + (21.2 + 65.3i)T + (-2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-34.9 - 48.1i)T + (-1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 - 69.3T + 4.48e3T^{2} \) |
| 71 | \( 1 + (-30.5 + 22.2i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-51.1 + 16.6i)T + (4.31e3 - 3.13e3i)T^{2} \) |
| 79 | \( 1 + (-22.3 + 30.7i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (7.17 + 9.87i)T + (-2.12e3 + 6.55e3i)T^{2} \) |
| 89 | \( 1 + 142.T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-117. - 85.2i)T + (2.90e3 + 8.94e3i)T^{2} \) |
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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
−11.36060680181217279739855676399, −10.06463524569294826967975816896, −9.026907479750438319632439428607, −8.130228131312006266532004241931, −7.40532317224549706138825374336, −6.14823548079282978592107993523, −5.51514896596791598195861487255, −3.40923971979236300856925690732, −2.17944465831674507107583597655, −0.843123155722498049557977543464,
2.35255589978283246621266265526, 3.57324083018238137537707847429, 4.59358705043631535498988003844, 5.69773680711789072359225701323, 6.96497295491720549317683799477, 8.251375802606320457709311317930, 9.421356297797713780596619645609, 9.863921998872423030380939795922, 10.55587124713010658116217235202, 11.56557517670463834435494037877
