| L(s) = 1 | + (−0.164 − 0.0132i)2-s + (0.845 + 0.534i)3-s + (−1.94 − 0.316i)4-s + (2.22 + 1.96i)5-s + (−0.131 − 0.0989i)6-s + (−0.158 + 0.164i)7-s + (0.635 + 0.156i)8-s + (0.428 + 0.903i)9-s + (−0.338 − 0.352i)10-s + (−2.22 + 4.68i)11-s + (−1.47 − 1.30i)12-s + (−3.60 − 0.160i)13-s + (0.0281 − 0.0249i)14-s + (0.826 + 2.85i)15-s + (3.64 + 1.21i)16-s + (−1.50 + 1.56i)17-s + ⋯ |
| L(s) = 1 | + (−0.116 − 0.00936i)2-s + (0.487 + 0.308i)3-s + (−0.973 − 0.158i)4-s + (0.994 + 0.880i)5-s + (−0.0537 − 0.0403i)6-s + (−0.0597 + 0.0621i)7-s + (0.224 + 0.0553i)8-s + (0.142 + 0.301i)9-s + (−0.107 − 0.111i)10-s + (−0.670 + 1.41i)11-s + (−0.426 − 0.377i)12-s + (−0.999 − 0.0443i)13-s + (0.00751 − 0.00665i)14-s + (0.213 + 0.736i)15-s + (0.910 + 0.303i)16-s + (−0.364 + 0.379i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.143 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.143 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.827246 + 0.955433i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.827246 + 0.955433i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.845 - 0.534i)T \) |
| 13 | \( 1 + (3.60 + 0.160i)T \) |
| good | 2 | \( 1 + (0.164 + 0.0132i)T + (1.97 + 0.320i)T^{2} \) |
| 5 | \( 1 + (-2.22 - 1.96i)T + (0.602 + 4.96i)T^{2} \) |
| 7 | \( 1 + (0.158 - 0.164i)T + (-0.281 - 6.99i)T^{2} \) |
| 11 | \( 1 + (2.22 - 4.68i)T + (-6.95 - 8.52i)T^{2} \) |
| 17 | \( 1 + (1.50 - 1.56i)T + (-0.684 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-2.38 + 4.13i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.04 - 1.81i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.67 - 0.458i)T + (28.6 + 4.65i)T^{2} \) |
| 31 | \( 1 + (0.846 - 6.97i)T + (-30.0 - 7.41i)T^{2} \) |
| 37 | \( 1 + (10.5 - 4.48i)T + (25.6 - 26.6i)T^{2} \) |
| 41 | \( 1 + (0.450 + 0.284i)T + (17.5 + 37.0i)T^{2} \) |
| 43 | \( 1 + (-3.19 - 1.35i)T + (29.7 + 31.0i)T^{2} \) |
| 47 | \( 1 + (-0.481 + 1.26i)T + (-35.1 - 31.1i)T^{2} \) |
| 53 | \( 1 + (-7.73 - 1.90i)T + (46.9 + 24.6i)T^{2} \) |
| 59 | \( 1 + (-8.17 + 2.72i)T + (47.1 - 35.4i)T^{2} \) |
| 61 | \( 1 + (-0.807 + 2.78i)T + (-51.5 - 32.6i)T^{2} \) |
| 67 | \( 1 + (-14.6 + 2.37i)T + (63.5 - 21.2i)T^{2} \) |
| 71 | \( 1 + (-0.600 + 14.8i)T + (-70.7 - 5.71i)T^{2} \) |
| 73 | \( 1 + (4.24 + 6.14i)T + (-25.8 + 68.2i)T^{2} \) |
| 79 | \( 1 + (-3.43 + 9.06i)T + (-59.1 - 52.3i)T^{2} \) |
| 83 | \( 1 + (13.0 + 6.86i)T + (47.1 + 68.3i)T^{2} \) |
| 89 | \( 1 + (-3.64 - 6.31i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.65 - 13.0i)T + (-89.2 + 38.0i)T^{2} \) |
| 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
−10.59436451643592922957550424493, −10.16201608746184419608202456187, −9.535606279861345973466933647521, −8.736129990699720549325032591836, −7.51301294705584200241782937015, −6.71952435022455118375619374419, −5.24032624395498879105324194974, −4.68425820872793806605355807094, −3.12183688976186189509702274200, −2.04266712301366389545408345623,
0.76385840225329433135219204053, 2.45591847638019696347119629287, 3.83961002827845736508677708043, 5.16733375625855659332803919000, 5.69518442523093332759195052691, 7.20628425784043717436542918731, 8.371480024947177746030809503552, 8.721071673501670854347724665281, 9.718214459639571750079933177260, 10.23563400762068755182749286668
