| L(s) = 1 | + (0.173 − 0.984i)2-s + (0.266 + 0.223i)3-s + (−0.939 − 0.342i)4-s + (−1.87 + 0.684i)5-s + (0.266 − 0.223i)6-s + (0.879 + 1.52i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 2.83i)9-s + (0.347 + 1.96i)10-s + (−2.11 + 3.66i)11-s + (−0.173 − 0.300i)12-s + (0.815 − 0.684i)13-s + (1.65 − 0.601i)14-s + (−0.652 − 0.237i)15-s + (0.766 + 0.642i)16-s + (1.23 − 7.02i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (0.153 + 0.128i)3-s + (−0.469 − 0.171i)4-s + (−0.840 + 0.305i)5-s + (0.108 − 0.0911i)6-s + (0.332 + 0.575i)7-s + (−0.176 + 0.306i)8-s + (−0.166 − 0.945i)9-s + (0.109 + 0.622i)10-s + (−0.637 + 1.10i)11-s + (−0.0501 − 0.0868i)12-s + (0.226 − 0.189i)13-s + (0.441 − 0.160i)14-s + (−0.168 − 0.0613i)15-s + (0.191 + 0.160i)16-s + (0.300 − 1.70i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.706920 - 0.226773i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.706920 - 0.226773i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 19 | \( 1 + (-3.93 + 1.86i)T \) |
| good | 3 | \( 1 + (-0.266 - 0.223i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (1.87 - 0.684i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.879 - 1.52i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.11 - 3.66i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.815 + 0.684i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.23 + 7.02i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (3.53 + 1.28i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.10 - 6.27i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-4.41 - 7.64i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6.45T + 37T^{2} \) |
| 41 | \( 1 + (-1.43 - 1.20i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (3.47 - 1.26i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.638 + 3.61i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-9.29 - 3.38i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (1.26 - 7.18i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (4.98 + 1.81i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (2.02 + 11.4i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (2.65 - 0.965i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-0.607 - 0.509i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (5.12 + 4.30i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.754 + 1.30i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-9.12 + 7.65i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-0.326 + 1.85i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−15.88628354657127170100774774232, −15.11089702845859496438868351328, −13.92758744846687959552924807342, −12.22888813486109849538623143032, −11.72468166047698521414173334376, −10.16245882802351460030199982301, −8.879727286427702875298729000581, −7.27962674212362147763732574797, −4.99195111483558045835725903285, −3.12877039704086847121502578657,
4.01573848747841797119004742400, 5.77604650880201637022251979632, 7.84351709393095974874095945906, 8.234523413196251755578946876219, 10.38679283850002009623447221747, 11.72603037984931372648160222508, 13.29866031875145859823242878999, 14.07190286841063755634598122640, 15.51460364656326312374347265048, 16.36223903460666929728197110099
