| L(s) = 1 | + (2.41 + 0.364i)2-s + (1.41 − 0.962i)3-s + (3.80 + 1.17i)4-s + (−0.219 − 2.93i)5-s + (3.76 − 1.81i)6-s + (−2.56 + 0.650i)7-s + (4.36 + 2.10i)8-s + (−0.0301 + 0.0769i)9-s + (0.537 − 7.16i)10-s + (−0.641 − 1.63i)11-s + (6.49 − 2.00i)12-s + (0.623 − 0.781i)13-s + (−6.43 + 0.639i)14-s + (−3.12 − 3.92i)15-s + (3.20 + 2.18i)16-s + (5.43 + 5.04i)17-s + ⋯ |
| L(s) = 1 | + (1.70 + 0.257i)2-s + (0.814 − 0.555i)3-s + (1.90 + 0.586i)4-s + (−0.0982 − 1.31i)5-s + (1.53 − 0.739i)6-s + (−0.969 + 0.245i)7-s + (1.54 + 0.743i)8-s + (−0.0100 + 0.0256i)9-s + (0.169 − 2.26i)10-s + (−0.193 − 0.492i)11-s + (1.87 − 0.578i)12-s + (0.172 − 0.216i)13-s + (−1.72 + 0.170i)14-s + (−0.808 − 1.01i)15-s + (0.802 + 0.546i)16-s + (1.31 + 1.22i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.28578 - 1.16028i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.28578 - 1.16028i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.56 - 0.650i)T \) |
| 13 | \( 1 + (-0.623 + 0.781i)T \) |
| good | 2 | \( 1 + (-2.41 - 0.364i)T + (1.91 + 0.589i)T^{2} \) |
| 3 | \( 1 + (-1.41 + 0.962i)T + (1.09 - 2.79i)T^{2} \) |
| 5 | \( 1 + (0.219 + 2.93i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (0.641 + 1.63i)T + (-8.06 + 7.48i)T^{2} \) |
| 17 | \( 1 + (-5.43 - 5.04i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.40 - 2.42i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.33 - 1.24i)T + (1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (1.47 - 6.45i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-2.21 + 3.84i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (7.47 - 2.30i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (6.02 + 2.89i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-1.01 + 0.489i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-2.71 - 0.408i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (11.8 + 3.64i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.733 + 9.78i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (5.94 - 1.83i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-3.92 + 6.79i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.18 - 5.18i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-4.49 + 0.676i)T + (69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (-6.91 - 11.9i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.08 - 6.37i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-3.45 + 8.80i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 - 1.33T + 97T^{2} \) |
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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
−10.72147152379272827611266887344, −9.498656864090573213376284240926, −8.415871381074369924726909710981, −7.87282460569477001335757262601, −6.69390631072528507369468505728, −5.67591148950410141308373924508, −5.15621580474263116106231816698, −3.70661695163040308799727834574, −3.18414812711952890545162048396, −1.69632526428597828432860434933,
2.56484267597891996097241196602, 3.15724796925133804856894873008, 3.73850620615419468138015302381, 4.87422144239797699294035672454, 6.09988170500046790640194312723, 6.82255928755487076656071391291, 7.61001395895824658642023927789, 9.241607045680802930884604280744, 10.03625738357854561212830685815, 10.68376590589865184758011522606
