| L(s) = 1 | + (7.21 + 7.21i)2-s + (−2.36 − 5.72i)3-s + 72.2i·4-s + (34.6 − 14.3i)5-s + (24.1 − 58.4i)6-s + (−123. − 51.2i)7-s + (−290. + 290. i)8-s + (144. − 144. i)9-s + (354. + 146. i)10-s + (205. − 496. i)11-s + (413. − 171. i)12-s + 864. i·13-s + (−523. − 1.26e3i)14-s + (−164. − 164. i)15-s − 1.88e3·16-s + (−1.17e3 − 194. i)17-s + ⋯ |
| L(s) = 1 | + (1.27 + 1.27i)2-s + (−0.152 − 0.366i)3-s + 2.25i·4-s + (0.620 − 0.257i)5-s + (0.274 − 0.662i)6-s + (−0.954 − 0.395i)7-s + (−1.60 + 1.60i)8-s + (0.595 − 0.595i)9-s + (1.12 + 0.463i)10-s + (0.512 − 1.23i)11-s + (0.828 − 0.343i)12-s + 1.41i·13-s + (−0.713 − 1.72i)14-s + (−0.188 − 0.188i)15-s − 1.83·16-s + (−0.986 − 0.162i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.82950 + 1.43650i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.82950 + 1.43650i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (1.17e3 + 194. i)T \) |
| good | 2 | \( 1 + (-7.21 - 7.21i)T + 32iT^{2} \) |
| 3 | \( 1 + (2.36 + 5.72i)T + (-171. + 171. i)T^{2} \) |
| 5 | \( 1 + (-34.6 + 14.3i)T + (2.20e3 - 2.20e3i)T^{2} \) |
| 7 | \( 1 + (123. + 51.2i)T + (1.18e4 + 1.18e4i)T^{2} \) |
| 11 | \( 1 + (-205. + 496. i)T + (-1.13e5 - 1.13e5i)T^{2} \) |
| 13 | \( 1 - 864. iT - 3.71e5T^{2} \) |
| 19 | \( 1 + (719. + 719. i)T + 2.47e6iT^{2} \) |
| 23 | \( 1 + (1.35e3 - 3.27e3i)T + (-4.55e6 - 4.55e6i)T^{2} \) |
| 29 | \( 1 + (-4.12e3 + 1.70e3i)T + (1.45e7 - 1.45e7i)T^{2} \) |
| 31 | \( 1 + (-758. - 1.83e3i)T + (-2.02e7 + 2.02e7i)T^{2} \) |
| 37 | \( 1 + (-881. - 2.12e3i)T + (-4.90e7 + 4.90e7i)T^{2} \) |
| 41 | \( 1 + (-3.64e3 - 1.51e3i)T + (8.19e7 + 8.19e7i)T^{2} \) |
| 43 | \( 1 + (-1.14e4 + 1.14e4i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + 1.33e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + (-2.22e4 - 2.22e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + (-1.47e3 + 1.47e3i)T - 7.14e8iT^{2} \) |
| 61 | \( 1 + (1.30e4 + 5.41e3i)T + (5.97e8 + 5.97e8i)T^{2} \) |
| 67 | \( 1 + 3.58e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + (8.87e3 + 2.14e4i)T + (-1.27e9 + 1.27e9i)T^{2} \) |
| 73 | \( 1 + (2.97e4 - 1.23e4i)T + (1.46e9 - 1.46e9i)T^{2} \) |
| 79 | \( 1 + (-1.33e4 + 3.21e4i)T + (-2.17e9 - 2.17e9i)T^{2} \) |
| 83 | \( 1 + (-2.45e4 - 2.45e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 - 1.17e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-1.19e5 + 4.94e4i)T + (6.07e9 - 6.07e9i)T^{2} \) |
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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
−17.52672818195678085409973300368, −16.52255151618560823603416148349, −15.55295739955309319706064782590, −13.78870270208358231940324741619, −13.38352747936725646204857940779, −11.90549923142596013182751545865, −9.135295270215763991379039077886, −6.91840821163651680191649794345, −6.10137842633521763279216667864, −3.97180590837959054742425398262,
2.38623356174467201897087480628, 4.41806109512296148103278660750, 6.14562116784934006134974605950, 9.861925600486132439144123959134, 10.55345518886104364409825611000, 12.43043689221761250580381737923, 13.10579362763479356757178999253, 14.60888262174693395459754587491, 15.77165882312335538400873113797, 17.90416969765369243881700313067
