| 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
−11.37656089362511160672840485250, −10.19815440706644357511262683512, −9.642698787224637831547567869615, −8.785514910007004888081292732474, −7.61044076060740883284509262512, −6.25272810703231561624861099456, −5.71256240648486689821759594495, −3.89947530389731320961931441147, −3.01416605096438937218833498995, −2.40430615216860869859207848140,
1.38123759981097948775621202023, 3.26892475659701790097431772230, 4.07525903551322820667774907700, 5.34495401161715748953706592578, 6.46962625141801584247074945189, 7.47721044558005543410116733122, 8.401384399221908740704123058163, 8.951566375044133602070867218026, 10.09039832057265990630957397714, 11.19365085267679429131998976897
