| L(s) = 1 | + (0.984 − 0.173i)2-s + (−0.0904 + 1.72i)3-s + (0.939 − 0.342i)4-s + (1.22 − 0.445i)5-s + (0.211 + 1.71i)6-s + (2.52 − 0.778i)7-s + (0.866 − 0.5i)8-s + (−2.98 − 0.312i)9-s + (1.12 − 0.651i)10-s + (0.276 − 0.759i)11-s + (0.506 + 1.65i)12-s + (0.540 + 1.48i)13-s + (2.35 − 1.20i)14-s + (0.660 + 2.15i)15-s + (0.766 − 0.642i)16-s + (0.739 + 1.28i)17-s + ⋯ |
| L(s) = 1 | + (0.696 − 0.122i)2-s + (−0.0521 + 0.998i)3-s + (0.469 − 0.171i)4-s + (0.547 − 0.199i)5-s + (0.0862 + 0.701i)6-s + (0.955 − 0.294i)7-s + (0.306 − 0.176i)8-s + (−0.994 − 0.104i)9-s + (0.356 − 0.206i)10-s + (0.0833 − 0.228i)11-s + (0.146 + 0.478i)12-s + (0.149 + 0.411i)13-s + (0.629 − 0.322i)14-s + (0.170 + 0.557i)15-s + (0.191 − 0.160i)16-s + (0.179 + 0.310i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.480i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.18647 + 0.560133i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.18647 + 0.560133i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.984 + 0.173i)T \) |
| 3 | \( 1 + (0.0904 - 1.72i)T \) |
| 7 | \( 1 + (-2.52 + 0.778i)T \) |
| good | 5 | \( 1 + (-1.22 + 0.445i)T + (3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (-0.276 + 0.759i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.540 - 1.48i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.739 - 1.28i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.492 - 0.284i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (6.98 + 1.23i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.907 - 2.49i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.00 - 2.75i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 - 2.55T + 37T^{2} \) |
| 41 | \( 1 + (-4.20 + 1.53i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1.31 + 7.47i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (2.29 + 0.835i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (7.73 + 4.46i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (10.7 + 9.05i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-2.35 + 6.47i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.97 - 11.2i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (11.3 + 6.56i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 11.9iT - 73T^{2} \) |
| 79 | \( 1 + (2.03 + 11.5i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-6.99 - 2.54i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-0.961 + 1.66i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-15.6 + 2.76i)T + (91.1 - 33.1i)T^{2} \) |
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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
−11.39912251256232165714976173595, −10.61104802887602166551236450838, −9.823104938423567541333887074679, −8.782154649465882433395808709624, −7.77528681487327174794931948934, −6.26171139922057443706797735188, −5.40323552457939256579323290911, −4.49915661511524648591299548355, −3.55508386866243247426079127116, −1.90657669921150947895380252635,
1.68754746194026001109163054258, 2.76060166019438721311019202450, 4.46527594127107122032334617043, 5.73089467894249991595084702817, 6.23214988316650234115129911487, 7.61345389785279524844773418346, 8.052510510843349048521252256512, 9.425020483888568405207857702136, 10.66232852282785744077318567675, 11.63663243715359665876089600666
