| L(s) = 1 | + (0.766 + 0.642i)2-s + (1.64 − 0.599i)3-s + (0.173 + 0.984i)4-s + (0.477 − 2.70i)5-s + (1.64 + 0.599i)6-s + (−1.91 − 3.32i)7-s + (−0.500 + 0.866i)8-s + (0.0532 − 0.0446i)9-s + (2.10 − 1.76i)10-s + (0.5 − 0.866i)11-s + (0.875 + 1.51i)12-s + (2.81 + 1.02i)13-s + (0.665 − 3.77i)14-s + (−0.836 − 4.74i)15-s + (−0.939 + 0.342i)16-s + (5.31 + 4.46i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (0.950 − 0.345i)3-s + (0.0868 + 0.492i)4-s + (0.213 − 1.21i)5-s + (0.672 + 0.244i)6-s + (−0.724 − 1.25i)7-s + (−0.176 + 0.306i)8-s + (0.0177 − 0.0148i)9-s + (0.666 − 0.558i)10-s + (0.150 − 0.261i)11-s + (0.252 + 0.437i)12-s + (0.781 + 0.284i)13-s + (0.177 − 1.00i)14-s + (−0.216 − 1.22i)15-s + (−0.234 + 0.0855i)16-s + (1.29 + 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.28753 - 0.577858i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.28753 - 0.577858i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (2.84 + 3.30i)T \) |
| good | 3 | \( 1 + (-1.64 + 0.599i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (-0.477 + 2.70i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (1.91 + 3.32i)T + (-3.5 + 6.06i)T^{2} \) |
| 13 | \( 1 + (-2.81 - 1.02i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-5.31 - 4.46i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.183 - 1.04i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (5.34 - 4.48i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.70 - 4.68i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 5.74T + 37T^{2} \) |
| 41 | \( 1 + (5.91 - 2.15i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.915 - 5.19i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-7.47 + 6.27i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (1.31 + 7.44i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-0.760 - 0.638i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.09 - 6.20i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.52 + 1.28i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.32 + 13.1i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (8.03 - 2.92i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (9.03 - 3.28i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-4.93 - 8.54i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (12.3 + 4.51i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (4.67 + 3.91i)T + (16.8 + 95.5i)T^{2} \) |
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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
−11.19365085267679429131998976897, −10.09039832057265990630957397714, −8.951566375044133602070867218026, −8.401384399221908740704123058163, −7.47721044558005543410116733122, −6.46962625141801584247074945189, −5.34495401161715748953706592578, −4.07525903551322820667774907700, −3.26892475659701790097431772230, −1.38123759981097948775621202023,
2.40430615216860869859207848140, 3.01416605096438937218833498995, 3.89947530389731320961931441147, 5.71256240648486689821759594495, 6.25272810703231561624861099456, 7.61044076060740883284509262512, 8.785514910007004888081292732474, 9.642698787224637831547567869615, 10.19815440706644357511262683512, 11.37656089362511160672840485250
