| L(s) = 1 | + (−0.427 + 1.34i)2-s + (1.64 − 0.541i)3-s + (−1.63 − 1.15i)4-s + (2.38 + 2.71i)5-s + (0.0266 + 2.44i)6-s + (3.65 + 0.480i)7-s + (2.25 − 1.71i)8-s + (2.41 − 1.78i)9-s + (−4.67 + 2.04i)10-s + (−1.82 − 3.69i)11-s + (−3.31 − 1.01i)12-s + (1.01 + 2.99i)13-s + (−2.20 + 4.71i)14-s + (5.38 + 3.17i)15-s + (1.34 + 3.76i)16-s + (−4.41 − 4.41i)17-s + ⋯ |
| L(s) = 1 | + (−0.302 + 0.953i)2-s + (0.949 − 0.312i)3-s + (−0.817 − 0.576i)4-s + (1.06 + 1.21i)5-s + (0.0108 + 0.999i)6-s + (1.37 + 0.181i)7-s + (0.796 − 0.604i)8-s + (0.804 − 0.594i)9-s + (−1.47 + 0.647i)10-s + (−0.549 − 1.11i)11-s + (−0.956 − 0.291i)12-s + (0.281 + 0.830i)13-s + (−0.590 + 1.26i)14-s + (1.39 + 0.820i)15-s + (0.335 + 0.942i)16-s + (−1.07 − 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.76760 + 1.13104i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.76760 + 1.13104i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.427 - 1.34i)T \) |
| 3 | \( 1 + (-1.64 + 0.541i)T \) |
| good | 5 | \( 1 + (-2.38 - 2.71i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (-3.65 - 0.480i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (1.82 + 3.69i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (-1.01 - 2.99i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (4.41 + 4.41i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.330 + 1.66i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (0.580 + 4.41i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (4.75 - 0.311i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (4.26 - 2.46i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.25 - 6.31i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (0.274 + 2.08i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (5.84 - 2.88i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (-0.735 + 0.197i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (6.09 - 9.12i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (-5.64 - 6.43i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (0.314 + 4.80i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (0.486 + 0.240i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (-1.08 - 2.61i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (5.99 - 14.4i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-1.74 - 6.51i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (8.46 + 7.42i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (-5.32 + 2.20i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (5.66 + 3.27i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.77696213771031813294716720155, −9.724585019025686905308846642554, −8.886093186547346261683009407249, −8.296028793441533887115551562631, −7.21742014220865634344743432113, −6.63975086295369254491471333631, −5.60293138464007455733497193375, −4.45576946904634718866123883553, −2.83714311180007339454493222675, −1.72182493524267429900943748338,
1.70526145792637680635760638946, 2.01527180257552950813136921805, 3.84581543710627974855380456804, 4.76490298894129378013484641521, 5.42387201381391588040483011788, 7.61402516418926402275959698626, 8.192111122547727426459978217921, 8.948579137826879528650085443996, 9.669620180658823920656246857358, 10.40094186944468288346109079819
