| L(s) = 1 | + (0.540 + 2.01i)2-s + (−8.21 + 14.2i)3-s + (10.0 − 5.82i)4-s + (17.2 − 17.2i)5-s + (−33.1 − 8.87i)6-s + (−2.95 + 11.0i)7-s + (40.8 + 40.8i)8-s + (−94.3 − 163. i)9-s + (44.1 + 25.4i)10-s + (101. − 27.2i)11-s + 191. i·12-s + (−166. − 27.5i)13-s − 23.7·14-s + (103. + 387. i)15-s + (32.8 − 56.9i)16-s + (65.7 − 37.9i)17-s + ⋯ |
| L(s) = 1 | + (0.135 + 0.504i)2-s + (−0.912 + 1.58i)3-s + (0.630 − 0.363i)4-s + (0.690 − 0.690i)5-s + (−0.920 − 0.246i)6-s + (−0.0602 + 0.224i)7-s + (0.637 + 0.637i)8-s + (−1.16 − 2.01i)9-s + (0.441 + 0.254i)10-s + (0.840 − 0.225i)11-s + 1.32i·12-s + (−0.986 − 0.163i)13-s − 0.121·14-s + (0.460 + 1.72i)15-s + (0.128 − 0.222i)16-s + (0.227 − 0.131i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 - 0.979i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.874347 + 0.713094i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.874347 + 0.713094i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (166. + 27.5i)T \) |
| good | 2 | \( 1 + (-0.540 - 2.01i)T + (-13.8 + 8i)T^{2} \) |
| 3 | \( 1 + (8.21 - 14.2i)T + (-40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (-17.2 + 17.2i)T - 625iT^{2} \) |
| 7 | \( 1 + (2.95 - 11.0i)T + (-2.07e3 - 1.20e3i)T^{2} \) |
| 11 | \( 1 + (-101. + 27.2i)T + (1.26e4 - 7.32e3i)T^{2} \) |
| 17 | \( 1 + (-65.7 + 37.9i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (313. + 84.0i)T + (1.12e5 + 6.51e4i)T^{2} \) |
| 23 | \( 1 + (174. + 100. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-432. + 748. i)T + (-3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 + (767. - 767. i)T - 9.23e5iT^{2} \) |
| 37 | \( 1 + (756. - 202. i)T + (1.62e6 - 9.37e5i)T^{2} \) |
| 41 | \( 1 + (-308. - 1.15e3i)T + (-2.44e6 + 1.41e6i)T^{2} \) |
| 43 | \( 1 + (-318. + 183. i)T + (1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (308. + 308. i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + 2.12e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (1.46e3 - 5.48e3i)T + (-1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (2.85e3 + 4.93e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (525. + 1.96e3i)T + (-1.74e7 + 1.00e7i)T^{2} \) |
| 71 | \( 1 + (-1.51e3 - 404. i)T + (2.20e7 + 1.27e7i)T^{2} \) |
| 73 | \( 1 + (1.57e3 + 1.57e3i)T + 2.83e7iT^{2} \) |
| 79 | \( 1 - 4.44e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + (4.02e3 - 4.02e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + (-1.06e4 + 2.84e3i)T + (5.43e7 - 3.13e7i)T^{2} \) |
| 97 | \( 1 + (-1.65e4 - 4.44e3i)T + (7.66e7 + 4.42e7i)T^{2} \) |
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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
−19.83644591926455842809686550483, −17.25233112876021800161060602507, −16.73384681997925912619247158058, −15.57454680097728349239147092060, −14.48067049180093776696233150504, −11.98376219291088216615618937899, −10.55186080402038262078745605636, −9.307366467293753463033818686483, −6.13372198019449794045094056781, −4.89597969184485138270516817094,
2.00729610404056295788149045289, 6.34118575478990446880202059557, 7.34030654587919998860237464228, 10.56023016012891677312239992932, 11.87378830002541878809540646218, 12.78741550070334309428480591621, 14.24219869662379485214340282160, 16.77039680424137194139271730442, 17.56630083862672651920835367388, 18.90826986056104873850189907219
