| L(s) = 1 | + (−0.654 − 0.291i)2-s + (0.0273 − 0.259i)3-s + (−0.994 − 1.10i)4-s + 4.00·5-s + (−0.0935 + 0.162i)6-s + (0.155 + 0.172i)7-s + (0.771 + 2.37i)8-s + (2.86 + 0.609i)9-s + (−2.62 − 1.16i)10-s + (1.14 − 0.244i)11-s + (−0.314 + 0.228i)12-s + (−3.10 + 1.83i)13-s + (−0.0513 − 0.158i)14-s + (0.109 − 1.04i)15-s + (−0.123 + 1.17i)16-s + (−0.0529 − 0.0112i)17-s + ⋯ |
| L(s) = 1 | + (−0.462 − 0.206i)2-s + (0.0157 − 0.149i)3-s + (−0.497 − 0.552i)4-s + 1.79·5-s + (−0.0381 + 0.0661i)6-s + (0.0586 + 0.0651i)7-s + (0.272 + 0.839i)8-s + (0.955 + 0.203i)9-s + (−0.828 − 0.368i)10-s + (0.346 − 0.0736i)11-s + (−0.0907 + 0.0659i)12-s + (−0.860 + 0.508i)13-s + (−0.0137 − 0.0422i)14-s + (0.0282 − 0.268i)15-s + (−0.0309 + 0.294i)16-s + (−0.0128 − 0.00272i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 + 0.586i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30617 - 0.423198i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30617 - 0.423198i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (3.10 - 1.83i)T \) |
| 31 | \( 1 + (5.56 - 0.125i)T \) |
| good | 2 | \( 1 + (0.654 + 0.291i)T + (1.33 + 1.48i)T^{2} \) |
| 3 | \( 1 + (-0.0273 + 0.259i)T + (-2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 - 4.00T + 5T^{2} \) |
| 7 | \( 1 + (-0.155 - 0.172i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (-1.14 + 0.244i)T + (10.0 - 4.47i)T^{2} \) |
| 17 | \( 1 + (0.0529 + 0.0112i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-0.263 - 2.50i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-5.76 + 6.40i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (3.71 + 1.65i)T + (19.4 + 21.5i)T^{2} \) |
| 37 | \( 1 + (-0.982 - 1.70i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.282 - 0.125i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (0.969 + 9.22i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (-3.22 - 2.34i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-2.30 - 7.09i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (9.43 - 4.19i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (7.02 - 12.1i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.69 + 8.12i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (6.05 + 1.28i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (-1.31 + 4.03i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (3.84 + 11.8i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (7.56 - 5.49i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (1.04 - 0.223i)T + (81.3 - 36.1i)T^{2} \) |
| 97 | \( 1 + (-12.3 - 13.7i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−10.57426340633083780979797251925, −10.32952268447975806264084064064, −9.262105920611795203723969069054, −9.005450975105005526847290533391, −7.39477234609228843373552957909, −6.35003288254903465218289074510, −5.40330937706697048396164561835, −4.50561055110511658478906075441, −2.31468787734592767722454278123, −1.41850447301509699640354161628,
1.48794419828166526246917122711, 3.11471369733441041778818511036, 4.61696878414234016410886584097, 5.55253106124425858608001791429, 6.86579285417925377652174504321, 7.51236808268422417384735798385, 9.012773791640809837392911133291, 9.494904710951564968651860063565, 10.00331682411661583619112328070, 11.10668651200898699344837024664
