| L(s) = 1 | + (0.104 − 0.994i)2-s + (−0.0185 − 1.73i)3-s + (−0.978 − 0.207i)4-s − 0.356·5-s + (−1.72 − 0.162i)6-s + (−0.655 − 2.01i)7-s + (−0.309 + 0.951i)8-s + (−2.99 + 0.0641i)9-s + (−0.0373 + 0.354i)10-s + (−1.13 − 0.240i)11-s + (−0.341 + 1.69i)12-s + (3.51 − 2.55i)13-s + (−2.07 + 0.441i)14-s + (0.00660 + 0.618i)15-s + (0.913 + 0.406i)16-s + (−1.85 − 2.06i)17-s + ⋯ |
| L(s) = 1 | + (0.0739 − 0.703i)2-s + (−0.0106 − 0.999i)3-s + (−0.489 − 0.103i)4-s − 0.159·5-s + (−0.703 − 0.0663i)6-s + (−0.247 − 0.762i)7-s + (−0.109 + 0.336i)8-s + (−0.999 + 0.0213i)9-s + (−0.0117 + 0.112i)10-s + (−0.341 − 0.0724i)11-s + (−0.0987 + 0.490i)12-s + (0.974 − 0.708i)13-s + (−0.554 + 0.117i)14-s + (0.00170 + 0.159i)15-s + (0.228 + 0.101i)16-s + (−0.449 − 0.499i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 - 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.231739 + 0.800576i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.231739 + 0.800576i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.104 + 0.994i)T \) |
| 3 | \( 1 + (0.0185 + 1.73i)T \) |
| 31 | \( 1 + (-4.58 + 3.15i)T \) |
| good | 5 | \( 1 + 0.356T + 5T^{2} \) |
| 7 | \( 1 + (0.655 + 2.01i)T + (-5.66 + 4.11i)T^{2} \) |
| 11 | \( 1 + (1.13 + 0.240i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-3.51 + 2.55i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (1.85 + 2.06i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (7.12 - 3.17i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (0.396 + 0.440i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (0.707 - 6.72i)T + (-28.3 - 6.02i)T^{2} \) |
| 37 | \( 1 + (-5.21 + 9.02i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-9.58 - 6.96i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (4.67 + 3.39i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (4.50 + 2.00i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-2.70 + 0.574i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (11.4 + 5.11i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (5.58 + 9.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 + (14.3 - 3.04i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (-4.20 + 4.67i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (-0.966 + 2.97i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-9.92 + 4.41i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (0.193 + 0.594i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.707 + 0.785i)T + (-10.1 - 96.4i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.65323827760719019186843838397, −9.410492404526234532668650240838, −8.333073076096673763626585025772, −7.74890210495971732250767571523, −6.55152242632313746185244631506, −5.75315921814916017691858646293, −4.29409649964899803434376930608, −3.22419334338147985314155738922, −1.93714526433660737213380010674, −0.45352214627957826528600634531,
2.55030672711018947959090931737, 4.00063852891726451440769250256, 4.63901770914004997533165425273, 5.99323545369174408730399723755, 6.36491377808429068665598609559, 7.973549728815579811222003514064, 8.699531567937800593563566673971, 9.313188796641981971920609293830, 10.32052288159484766901900544812, 11.19313059101088708209829698637
