| L(s) = 1 | + (−1.63 + 1.63i)2-s + (3.76 + 1.55i)3-s + 26.6i·4-s + (−7.23 + 17.4i)5-s + (−8.70 + 3.60i)6-s + (93.1 + 224. i)7-s + (−95.9 − 95.9i)8-s + (−160. − 160. i)9-s + (−16.7 − 40.3i)10-s + (522. − 216. i)11-s + (−41.5 + 100. i)12-s − 679. i·13-s + (−520. − 215. i)14-s + (−54.4 + 54.4i)15-s − 539.·16-s + (755. + 921. i)17-s + ⋯ |
| L(s) = 1 | + (−0.289 + 0.289i)2-s + (0.241 + 0.100i)3-s + 0.832i·4-s + (−0.129 + 0.312i)5-s + (−0.0987 + 0.0408i)6-s + (0.718 + 1.73i)7-s + (−0.529 − 0.529i)8-s + (−0.658 − 0.658i)9-s + (−0.0528 − 0.127i)10-s + (1.30 − 0.539i)11-s + (−0.0832 + 0.201i)12-s − 1.11i·13-s + (−0.709 − 0.293i)14-s + (−0.0624 + 0.0624i)15-s − 0.526·16-s + (0.634 + 0.773i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0519 - 0.998i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0519 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.900547 + 0.854892i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.900547 + 0.854892i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-755. - 921. i)T \) |
| good | 2 | \( 1 + (1.63 - 1.63i)T - 32iT^{2} \) |
| 3 | \( 1 + (-3.76 - 1.55i)T + (171. + 171. i)T^{2} \) |
| 5 | \( 1 + (7.23 - 17.4i)T + (-2.20e3 - 2.20e3i)T^{2} \) |
| 7 | \( 1 + (-93.1 - 224. i)T + (-1.18e4 + 1.18e4i)T^{2} \) |
| 11 | \( 1 + (-522. + 216. i)T + (1.13e5 - 1.13e5i)T^{2} \) |
| 13 | \( 1 + 679. iT - 3.71e5T^{2} \) |
| 19 | \( 1 + (185. - 185. i)T - 2.47e6iT^{2} \) |
| 23 | \( 1 + (-831. + 344. i)T + (4.55e6 - 4.55e6i)T^{2} \) |
| 29 | \( 1 + (-1.80e3 + 4.36e3i)T + (-1.45e7 - 1.45e7i)T^{2} \) |
| 31 | \( 1 + (-429. - 177. i)T + (2.02e7 + 2.02e7i)T^{2} \) |
| 37 | \( 1 + (8.36e3 + 3.46e3i)T + (4.90e7 + 4.90e7i)T^{2} \) |
| 41 | \( 1 + (-3.41e3 - 8.24e3i)T + (-8.19e7 + 8.19e7i)T^{2} \) |
| 43 | \( 1 + (-6.16e3 - 6.16e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + 1.66e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + (-273. + 273. i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 + (1.73e4 + 1.73e4i)T + 7.14e8iT^{2} \) |
| 61 | \( 1 + (3.93e3 + 9.49e3i)T + (-5.97e8 + 5.97e8i)T^{2} \) |
| 67 | \( 1 + 3.46e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + (5.47e4 + 2.26e4i)T + (1.27e9 + 1.27e9i)T^{2} \) |
| 73 | \( 1 + (-7.34e3 + 1.77e4i)T + (-1.46e9 - 1.46e9i)T^{2} \) |
| 79 | \( 1 + (5.67e4 - 2.35e4i)T + (2.17e9 - 2.17e9i)T^{2} \) |
| 83 | \( 1 + (-2.81e4 + 2.81e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 - 3.83e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-2.62e4 + 6.34e4i)T + (-6.07e9 - 6.07e9i)T^{2} \) |
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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
−17.98751035149878221719428986054, −17.06816783911650742531744219635, −15.39231794718018351209051908780, −14.60366459089765017818249265295, −12.44355353722031365547567341341, −11.54512858004659430874958092771, −9.048362528044611494437604400783, −8.223149874889027995905446919652, −6.04282028285723114324680332457, −3.17908535967026066141935789330,
1.29667002915910095458395299394, 4.62450110559272015044319845475, 7.08224354068160983829017315147, 8.971190318288145232080792714176, 10.51575231392569937223646697260, 11.66309542422083511742119304737, 14.05986916613707676505732589330, 14.31118433821173733751678227849, 16.53486974399362301167983938428, 17.47895753100805063925437055408
