| L(s) = 1 | + (−2.92 − 1.21i)2-s + (8.22 − 1.63i)3-s + (−4.21 − 4.21i)4-s + (21.2 − 31.7i)5-s + (−26.0 − 5.18i)6-s + (5.22 + 7.81i)7-s + (26.6 + 64.2i)8-s + (−9.85 + 4.08i)9-s + (−100. + 67.3i)10-s + (−30.2 + 152. i)11-s + (−41.5 − 27.7i)12-s + (144. − 144. i)13-s + (−5.80 − 29.2i)14-s + (122. − 296. i)15-s − 125. i·16-s + (−37.1 + 286. i)17-s + ⋯ |
| L(s) = 1 | + (−0.731 − 0.303i)2-s + (0.913 − 0.181i)3-s + (−0.263 − 0.263i)4-s + (0.849 − 1.27i)5-s + (−0.723 − 0.144i)6-s + (0.106 + 0.159i)7-s + (0.416 + 1.00i)8-s + (−0.121 + 0.0503i)9-s + (−1.00 + 0.673i)10-s + (−0.250 + 1.25i)11-s + (−0.288 − 0.192i)12-s + (0.854 − 0.854i)13-s + (−0.0296 − 0.149i)14-s + (0.545 − 1.31i)15-s − 0.488i·16-s + (−0.128 + 0.991i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.935415 - 0.573861i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.935415 - 0.573861i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (37.1 - 286. i)T \) |
| good | 2 | \( 1 + (2.92 + 1.21i)T + (11.3 + 11.3i)T^{2} \) |
| 3 | \( 1 + (-8.22 + 1.63i)T + (74.8 - 30.9i)T^{2} \) |
| 5 | \( 1 + (-21.2 + 31.7i)T + (-239. - 577. i)T^{2} \) |
| 7 | \( 1 + (-5.22 - 7.81i)T + (-918. + 2.21e3i)T^{2} \) |
| 11 | \( 1 + (30.2 - 152. i)T + (-1.35e4 - 5.60e3i)T^{2} \) |
| 13 | \( 1 + (-144. + 144. i)T - 2.85e4iT^{2} \) |
| 19 | \( 1 + (73.3 + 30.3i)T + (9.21e4 + 9.21e4i)T^{2} \) |
| 23 | \( 1 + (-411. - 81.9i)T + (2.58e5 + 1.07e5i)T^{2} \) |
| 29 | \( 1 + (-696. - 465. i)T + (2.70e5 + 6.53e5i)T^{2} \) |
| 31 | \( 1 + (60.5 + 304. i)T + (-8.53e5 + 3.53e5i)T^{2} \) |
| 37 | \( 1 + (2.14e3 - 427. i)T + (1.73e6 - 7.17e5i)T^{2} \) |
| 41 | \( 1 + (1.18e3 + 1.76e3i)T + (-1.08e6 + 2.61e6i)T^{2} \) |
| 43 | \( 1 + (-745. + 308. i)T + (2.41e6 - 2.41e6i)T^{2} \) |
| 47 | \( 1 + (1.06e3 - 1.06e3i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + (4.15e3 + 1.72e3i)T + (5.57e6 + 5.57e6i)T^{2} \) |
| 59 | \( 1 + (-2.14e3 - 5.17e3i)T + (-8.56e6 + 8.56e6i)T^{2} \) |
| 61 | \( 1 + (2.08e3 - 1.39e3i)T + (5.29e6 - 1.27e7i)T^{2} \) |
| 67 | \( 1 + 2.17e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + (-7.33e3 + 1.45e3i)T + (2.34e7 - 9.72e6i)T^{2} \) |
| 73 | \( 1 + (-99.3 + 148. i)T + (-1.08e7 - 2.62e7i)T^{2} \) |
| 79 | \( 1 + (-992. + 4.99e3i)T + (-3.59e7 - 1.49e7i)T^{2} \) |
| 83 | \( 1 + (1.00e3 - 2.43e3i)T + (-3.35e7 - 3.35e7i)T^{2} \) |
| 89 | \( 1 + (1.06e3 + 1.06e3i)T + 6.27e7iT^{2} \) |
| 97 | \( 1 + (1.10e3 + 737. i)T + (3.38e7 + 8.17e7i)T^{2} \) |
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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
−17.87361649063181971985519742315, −17.12609669097970875527253664242, −15.17170719482569351994235720379, −13.74702571449251205239575366939, −12.74857143387625214517078498284, −10.39449467109305055863512663905, −9.065260305592650229028730935699, −8.301588455293492708915073252973, −5.23629273498933196837327207964, −1.72134930603999571554360768941,
3.19177809838367154640788187562, 6.62903022672191526973728705584, 8.386076990033010289368786714185, 9.505201811250266530917687330417, 10.99536825352441309415213330418, 13.66311155864416853892279782267, 14.13272989192311592173095374669, 15.85644869351302795031046067407, 17.28463615086444415978240692357, 18.48449038495227445308131223121
