| L(s) = 1 | + (−0.707 + 0.376i)2-s + (−1.13 + 1.31i)3-s + (−0.759 + 1.12i)4-s + (1.77 + 2.84i)5-s + (0.307 − 1.35i)6-s + (1.17 + 4.10i)7-s + (0.281 − 2.67i)8-s + (−0.440 − 2.96i)9-s + (−2.32 − 1.34i)10-s + (1.99 + 2.65i)11-s + (−0.617 − 2.26i)12-s + (−0.675 − 4.80i)13-s + (−2.37 − 2.46i)14-s + (−5.73 − 0.885i)15-s + (−0.209 − 0.518i)16-s + (3.53 + 3.92i)17-s + ⋯ |
| L(s) = 1 | + (−0.500 + 0.266i)2-s + (−0.653 + 0.757i)3-s + (−0.379 + 0.562i)4-s + (0.794 + 1.27i)5-s + (0.125 − 0.552i)6-s + (0.445 + 1.55i)7-s + (0.0994 − 0.946i)8-s + (−0.146 − 0.989i)9-s + (−0.735 − 0.424i)10-s + (0.601 + 0.799i)11-s + (−0.178 − 0.655i)12-s + (−0.187 − 1.33i)13-s + (−0.635 − 0.658i)14-s + (−1.48 − 0.228i)15-s + (−0.0523 − 0.129i)16-s + (0.856 + 0.951i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0943234 + 0.841793i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0943234 + 0.841793i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.13 - 1.31i)T \) |
| 11 | \( 1 + (-1.99 - 2.65i)T \) |
| good | 2 | \( 1 + (0.707 - 0.376i)T + (1.11 - 1.65i)T^{2} \) |
| 5 | \( 1 + (-1.77 - 2.84i)T + (-2.19 + 4.49i)T^{2} \) |
| 7 | \( 1 + (-1.17 - 4.10i)T + (-5.93 + 3.70i)T^{2} \) |
| 13 | \( 1 + (0.675 + 4.80i)T + (-12.4 + 3.58i)T^{2} \) |
| 17 | \( 1 + (-3.53 - 3.92i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (1.55 + 0.163i)T + (18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (1.01 + 2.79i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.53 - 1.48i)T + (1.01 + 28.9i)T^{2} \) |
| 31 | \( 1 + (3.86 + 4.94i)T + (-7.49 + 30.0i)T^{2} \) |
| 37 | \( 1 + (1.10 + 10.5i)T + (-36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (-3.87 + 3.74i)T + (1.43 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-3.85 - 4.58i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-4.82 + 3.25i)T + (17.6 - 43.5i)T^{2} \) |
| 53 | \( 1 + (-4.45 + 1.44i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (0.230 - 3.29i)T + (-58.4 - 8.21i)T^{2} \) |
| 61 | \( 1 + (1.83 + 1.43i)T + (14.7 + 59.1i)T^{2} \) |
| 67 | \( 1 + (0.546 - 3.09i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (2.09 - 1.88i)T + (7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (0.100 - 0.225i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-5.33 - 10.0i)T + (-44.1 + 65.4i)T^{2} \) |
| 83 | \( 1 + (-5.35 - 0.752i)T + (79.7 + 22.8i)T^{2} \) |
| 89 | \( 1 + (13.5 - 7.79i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (10.2 + 6.39i)T + (42.5 + 87.1i)T^{2} \) |
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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
−12.32228732093254915406582103136, −11.00856253589352862788091789846, −10.19354702598989188850275629236, −9.477370945931004877840218507004, −8.585523903438288339202687099606, −7.36759542589033319443898014525, −6.13753596998535719561157678958, −5.47125894240831815378971903044, −3.89746366502031307511934040206, −2.55146260237257626956904697965,
0.910783763259474680693522327616, 1.58812622030980212185333449842, 4.43435878565846822805236120439, 5.22936863475777075382230578851, 6.31688259769264473864470707032, 7.50722177999468483745563997650, 8.623089289111345850680330903554, 9.471583734021112444941154860056, 10.37221007887943212976625764034, 11.30564159416137391098558662764
