| L(s) = 1 | + (−7.43 − 7.43i)2-s + (−9.60 − 23.1i)3-s + 78.4i·4-s + (51.4 − 21.3i)5-s + (−100. + 243. i)6-s + (−84.7 − 35.1i)7-s + (344. − 344. i)8-s + (−273. + 273. i)9-s + (−540. − 224. i)10-s + (−15.0 + 36.2i)11-s + (1.81e3 − 753. i)12-s − 243. i·13-s + (369. + 891. i)14-s + (−988. − 988. i)15-s − 2.61e3·16-s + (−952. + 716. i)17-s + ⋯ |
| L(s) = 1 | + (−1.31 − 1.31i)2-s + (−0.616 − 1.48i)3-s + 2.45i·4-s + (0.920 − 0.381i)5-s + (−1.14 + 2.76i)6-s + (−0.654 − 0.270i)7-s + (1.90 − 1.90i)8-s + (−1.12 + 1.12i)9-s + (−1.71 − 0.708i)10-s + (−0.0373 + 0.0902i)11-s + (3.64 − 1.51i)12-s − 0.399i·13-s + (0.503 + 1.21i)14-s + (−1.13 − 1.13i)15-s − 2.55·16-s + (−0.798 + 0.601i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.212143 + 0.387598i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.212143 + 0.387598i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (952. - 716. i)T \) |
| good | 2 | \( 1 + (7.43 + 7.43i)T + 32iT^{2} \) |
| 3 | \( 1 + (9.60 + 23.1i)T + (-171. + 171. i)T^{2} \) |
| 5 | \( 1 + (-51.4 + 21.3i)T + (2.20e3 - 2.20e3i)T^{2} \) |
| 7 | \( 1 + (84.7 + 35.1i)T + (1.18e4 + 1.18e4i)T^{2} \) |
| 11 | \( 1 + (15.0 - 36.2i)T + (-1.13e5 - 1.13e5i)T^{2} \) |
| 13 | \( 1 + 243. iT - 3.71e5T^{2} \) |
| 19 | \( 1 + (438. + 438. i)T + 2.47e6iT^{2} \) |
| 23 | \( 1 + (-1.43e3 + 3.46e3i)T + (-4.55e6 - 4.55e6i)T^{2} \) |
| 29 | \( 1 + (-4.22e3 + 1.74e3i)T + (1.45e7 - 1.45e7i)T^{2} \) |
| 31 | \( 1 + (2.55e3 + 6.15e3i)T + (-2.02e7 + 2.02e7i)T^{2} \) |
| 37 | \( 1 + (-1.16e3 - 2.81e3i)T + (-4.90e7 + 4.90e7i)T^{2} \) |
| 41 | \( 1 + (-6.25e3 - 2.59e3i)T + (8.19e7 + 8.19e7i)T^{2} \) |
| 43 | \( 1 + (1.13e4 - 1.13e4i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + 2.18e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + (3.36e3 + 3.36e3i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + (-2.75e4 + 2.75e4i)T - 7.14e8iT^{2} \) |
| 61 | \( 1 + (-1.12e4 - 4.66e3i)T + (5.97e8 + 5.97e8i)T^{2} \) |
| 67 | \( 1 - 4.30e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-2.24e4 - 5.42e4i)T + (-1.27e9 + 1.27e9i)T^{2} \) |
| 73 | \( 1 + (-4.41e4 + 1.82e4i)T + (1.46e9 - 1.46e9i)T^{2} \) |
| 79 | \( 1 + (-4.94e3 + 1.19e4i)T + (-2.17e9 - 2.17e9i)T^{2} \) |
| 83 | \( 1 + (4.69e4 + 4.69e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 + 7.29e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (7.73e4 - 3.20e4i)T + (6.07e9 - 6.07e9i)T^{2} \) |
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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
−17.47670974549418295135722444374, −16.77107328392796226554421394995, −13.15120437804361961646737458199, −12.80797821859452315897838868525, −11.33249477998398225428285500900, −9.950058229453250393422908562352, −8.363378761880317009523873842662, −6.63660143449037963916576025607, −2.16516043556819868978163070623, −0.56822408725602740817490121038,
5.36582089754752997482388691613, 6.60392426886094325742140352854, 9.084811838604564979512292118848, 9.822166805224341856338704850669, 10.89704279021640594993890850329, 14.14763054383501735590888515757, 15.49118667741117138592998586708, 16.15626030846233788500030368802, 17.22618653406742621264618992355, 18.06679814405436660315385178628
