| L(s) = 1 | + (3.81 + 3.81i)2-s + (6.88 + 16.6i)3-s − 2.90i·4-s + (−27.9 + 11.5i)5-s + (−37.1 + 89.6i)6-s + (62.5 + 25.9i)7-s + (133. − 133. i)8-s + (−57.0 + 57.0i)9-s + (−150. − 62.4i)10-s + (−26.4 + 63.8i)11-s + (48.3 − 20.0i)12-s − 898. i·13-s + (139. + 337. i)14-s + (−384. − 384. i)15-s + 922.·16-s + (−1.17e3 − 178. i)17-s + ⋯ |
| L(s) = 1 | + (0.674 + 0.674i)2-s + (0.441 + 1.06i)3-s − 0.0909i·4-s + (−0.500 + 0.207i)5-s + (−0.421 + 1.01i)6-s + (0.482 + 0.199i)7-s + (0.735 − 0.735i)8-s + (−0.234 + 0.234i)9-s + (−0.476 − 0.197i)10-s + (−0.0659 + 0.159i)11-s + (0.0969 − 0.0401i)12-s − 1.47i·13-s + (0.190 + 0.460i)14-s + (−0.441 − 0.441i)15-s + 0.900·16-s + (−0.988 − 0.150i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.224 - 0.974i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.56241 + 1.24322i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.56241 + 1.24322i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (1.17e3 + 178. i)T \) |
| good | 2 | \( 1 + (-3.81 - 3.81i)T + 32iT^{2} \) |
| 3 | \( 1 + (-6.88 - 16.6i)T + (-171. + 171. i)T^{2} \) |
| 5 | \( 1 + (27.9 - 11.5i)T + (2.20e3 - 2.20e3i)T^{2} \) |
| 7 | \( 1 + (-62.5 - 25.9i)T + (1.18e4 + 1.18e4i)T^{2} \) |
| 11 | \( 1 + (26.4 - 63.8i)T + (-1.13e5 - 1.13e5i)T^{2} \) |
| 13 | \( 1 + 898. iT - 3.71e5T^{2} \) |
| 19 | \( 1 + (-672. - 672. i)T + 2.47e6iT^{2} \) |
| 23 | \( 1 + (800. - 1.93e3i)T + (-4.55e6 - 4.55e6i)T^{2} \) |
| 29 | \( 1 + (5.52e3 - 2.28e3i)T + (1.45e7 - 1.45e7i)T^{2} \) |
| 31 | \( 1 + (2.71e3 + 6.54e3i)T + (-2.02e7 + 2.02e7i)T^{2} \) |
| 37 | \( 1 + (1.77e3 + 4.27e3i)T + (-4.90e7 + 4.90e7i)T^{2} \) |
| 41 | \( 1 + (-1.83e4 - 7.58e3i)T + (8.19e7 + 8.19e7i)T^{2} \) |
| 43 | \( 1 + (4.63e3 - 4.63e3i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 - 2.02e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + (-1.18e4 - 1.18e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + (-2.33e4 + 2.33e4i)T - 7.14e8iT^{2} \) |
| 61 | \( 1 + (-3.30e4 - 1.36e4i)T + (5.97e8 + 5.97e8i)T^{2} \) |
| 67 | \( 1 + 6.32e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-6.94e3 - 1.67e4i)T + (-1.27e9 + 1.27e9i)T^{2} \) |
| 73 | \( 1 + (-4.29e4 + 1.77e4i)T + (1.46e9 - 1.46e9i)T^{2} \) |
| 79 | \( 1 + (-2.58e3 + 6.24e3i)T + (-2.17e9 - 2.17e9i)T^{2} \) |
| 83 | \( 1 + (7.92e3 + 7.92e3i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 - 9.56e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (1.22e5 - 5.06e4i)T + (6.07e9 - 6.07e9i)T^{2} \) |
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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
−18.00429266883772331451522286451, −16.14741249177305617059209043332, −15.25241056451578888012858313493, −14.72886840777994611017249471332, −13.12375533154001150437278615172, −11.01143366098014719971467830180, −9.619557986825695943086406963964, −7.65572162049956543707354701976, −5.44444888936393060333257749193, −3.88432278906731284018646430132,
2.04224334361543558898955571120, 4.30664052715683496053642456273, 7.18617712610353819960514236162, 8.518525464424865567779182524918, 11.18553964846358120646744365924, 12.23162616606246055194927276312, 13.39840685191591864999787607352, 14.26814661975552008743040210857, 16.29645164659142161827250476214, 17.82090234409531294965213266287
