| 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
−18.06679814405436660315385178628, −17.22618653406742621264618992355, −16.15626030846233788500030368802, −15.49118667741117138592998586708, −14.14763054383501735590888515757, −10.89704279021640594993890850329, −9.822166805224341856338704850669, −9.084811838604564979512292118848, −6.60392426886094325742140352854, −5.36582089754752997482388691613,
0.56822408725602740817490121038, 2.16516043556819868978163070623, 6.63660143449037963916576025607, 8.363378761880317009523873842662, 9.950058229453250393422908562352, 11.33249477998398225428285500900, 12.80797821859452315897838868525, 13.15120437804361961646737458199, 16.77107328392796226554421394995, 17.47670974549418295135722444374
