| L(s) = 1 | + (1.30 − 1.51i)2-s + (−7.37 + 4.74i)3-s + (−0.569 − 3.95i)4-s + (−7.38 − 16.1i)5-s + (−2.49 + 17.3i)6-s + (−23.1 + 6.79i)7-s + (−6.73 − 4.32i)8-s + (20.7 − 45.3i)9-s + (−34.0 − 10.0i)10-s + (28.1 + 32.5i)11-s + (22.9 + 26.5i)12-s + (−18.1 − 5.31i)13-s + (−20.0 + 43.8i)14-s + (131. + 84.2i)15-s + (−15.3 + 4.50i)16-s + (−2.93 + 20.4i)17-s + ⋯ |
| L(s) = 1 | + (0.463 − 0.534i)2-s + (−1.41 + 0.912i)3-s + (−0.0711 − 0.494i)4-s + (−0.660 − 1.44i)5-s + (−0.169 + 1.18i)6-s + (−1.24 + 0.366i)7-s + (−0.297 − 0.191i)8-s + (0.767 − 1.68i)9-s + (−1.07 − 0.316i)10-s + (0.772 + 0.891i)11-s + (0.552 + 0.637i)12-s + (−0.386 − 0.113i)13-s + (−0.382 + 0.837i)14-s + (2.25 + 1.45i)15-s + (−0.239 + 0.0704i)16-s + (−0.0419 + 0.291i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.172i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0270398 - 0.311735i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0270398 - 0.311735i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.30 + 1.51i)T \) |
| 23 | \( 1 + (95.2 + 55.6i)T \) |
| good | 3 | \( 1 + (7.37 - 4.74i)T + (11.2 - 24.5i)T^{2} \) |
| 5 | \( 1 + (7.38 + 16.1i)T + (-81.8 + 94.4i)T^{2} \) |
| 7 | \( 1 + (23.1 - 6.79i)T + (288. - 185. i)T^{2} \) |
| 11 | \( 1 + (-28.1 - 32.5i)T + (-189. + 1.31e3i)T^{2} \) |
| 13 | \( 1 + (18.1 + 5.31i)T + (1.84e3 + 1.18e3i)T^{2} \) |
| 17 | \( 1 + (2.93 - 20.4i)T + (-4.71e3 - 1.38e3i)T^{2} \) |
| 19 | \( 1 + (18.5 + 128. i)T + (-6.58e3 + 1.93e3i)T^{2} \) |
| 29 | \( 1 + (-3.60 + 25.0i)T + (-2.34e4 - 6.87e3i)T^{2} \) |
| 31 | \( 1 + (4.65 + 2.98i)T + (1.23e4 + 2.70e4i)T^{2} \) |
| 37 | \( 1 + (-29.2 + 64.0i)T + (-3.31e4 - 3.82e4i)T^{2} \) |
| 41 | \( 1 + (-13.0 - 28.5i)T + (-4.51e4 + 5.20e4i)T^{2} \) |
| 43 | \( 1 + (-32.4 + 20.8i)T + (3.30e4 - 7.23e4i)T^{2} \) |
| 47 | \( 1 + 566.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-305. + 89.6i)T + (1.25e5 - 8.04e4i)T^{2} \) |
| 59 | \( 1 + (-359. - 105. i)T + (1.72e5 + 1.11e5i)T^{2} \) |
| 61 | \( 1 + (578. + 371. i)T + (9.42e4 + 2.06e5i)T^{2} \) |
| 67 | \( 1 + (-379. + 438. i)T + (-4.28e4 - 2.97e5i)T^{2} \) |
| 71 | \( 1 + (657. - 758. i)T + (-5.09e4 - 3.54e5i)T^{2} \) |
| 73 | \( 1 + (29.0 + 202. i)T + (-3.73e5 + 1.09e5i)T^{2} \) |
| 79 | \( 1 + (-691. - 202. i)T + (4.14e5 + 2.66e5i)T^{2} \) |
| 83 | \( 1 + (157. - 345. i)T + (-3.74e5 - 4.32e5i)T^{2} \) |
| 89 | \( 1 + (-682. + 438. i)T + (2.92e5 - 6.41e5i)T^{2} \) |
| 97 | \( 1 + (-69.5 - 152. i)T + (-5.97e5 + 6.89e5i)T^{2} \) |
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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
−15.17744698623033077612299411249, −12.91726167355475285085938970491, −12.29738387186628852943773089388, −11.48737852569483811906899778756, −9.978941610312373803970126458799, −9.138057401735030690041479199187, −6.45246874392326274203134262987, −5.02056409487211675342095442442, −4.12030698566665683325809784885, −0.24401952227312718281775973826,
3.58467641993129643498706562968, 6.03924618251528047506711715609, 6.62914405125781173722592133957, 7.62693725880218212242396093622, 10.23258242362515112576865324174, 11.46320481994579950909533612802, 12.21350835948743661788951988941, 13.49059070235761965775299576041, 14.59529156000625215738574202859, 16.13470765692389949038096962035
