| L(s) = 1 | + (−2.98 + 1.08i)2-s + (1.59 − 1.33i)4-s + (6.63 + 5.56i)5-s + (−4.36 − 7.55i)7-s + (9.39 − 16.2i)8-s + (−25.8 − 9.40i)10-s + (3.07 − 5.32i)11-s + (−4.94 − 28.0i)13-s + (21.2 + 17.8i)14-s + (−13.2 + 75.1i)16-s + (−69.5 + 25.3i)17-s + (67.3 − 48.2i)19-s + 18.0·20-s + (−3.39 + 19.2i)22-s + (132. − 111. i)23-s + ⋯ |
| L(s) = 1 | + (−1.05 + 0.383i)2-s + (0.199 − 0.167i)4-s + (0.593 + 0.497i)5-s + (−0.235 − 0.408i)7-s + (0.415 − 0.719i)8-s + (−0.816 − 0.297i)10-s + (0.0842 − 0.145i)11-s + (−0.105 − 0.598i)13-s + (0.405 + 0.340i)14-s + (−0.207 + 1.17i)16-s + (−0.992 + 0.361i)17-s + (0.813 − 0.582i)19-s + 0.201·20-s + (−0.0328 + 0.186i)22-s + (1.20 − 1.01i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.961755 + 0.0658991i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.961755 + 0.0658991i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + (-67.3 + 48.2i)T \) |
| good | 2 | \( 1 + (2.98 - 1.08i)T + (6.12 - 5.14i)T^{2} \) |
| 5 | \( 1 + (-6.63 - 5.56i)T + (21.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (4.36 + 7.55i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-3.07 + 5.32i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (4.94 + 28.0i)T + (-2.06e3 + 751. i)T^{2} \) |
| 17 | \( 1 + (69.5 - 25.3i)T + (3.76e3 - 3.15e3i)T^{2} \) |
| 23 | \( 1 + (-132. + 111. i)T + (2.11e3 - 1.19e4i)T^{2} \) |
| 29 | \( 1 + (-276. - 100. i)T + (1.86e4 + 1.56e4i)T^{2} \) |
| 31 | \( 1 + (-129. - 224. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 81.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-16.2 + 91.9i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 + (-109. - 91.5i)T + (1.38e4 + 7.82e4i)T^{2} \) |
| 47 | \( 1 + (-295. - 107. i)T + (7.95e4 + 6.67e4i)T^{2} \) |
| 53 | \( 1 + (82.5 - 69.2i)T + (2.58e4 - 1.46e5i)T^{2} \) |
| 59 | \( 1 + (179. - 65.2i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-234. + 196. i)T + (3.94e4 - 2.23e5i)T^{2} \) |
| 67 | \( 1 + (171. + 62.4i)T + (2.30e5 + 1.93e5i)T^{2} \) |
| 71 | \( 1 + (511. + 429. i)T + (6.21e4 + 3.52e5i)T^{2} \) |
| 73 | \( 1 + (-179. + 1.01e3i)T + (-3.65e5 - 1.33e5i)T^{2} \) |
| 79 | \( 1 + (177. - 1.00e3i)T + (-4.63e5 - 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-316. - 547. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (227. + 1.29e3i)T + (-6.62e5 + 2.41e5i)T^{2} \) |
| 97 | \( 1 + (546. - 198. i)T + (6.99e5 - 5.86e5i)T^{2} \) |
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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
−12.38953280368022229977906688846, −10.73478530960684399982840068428, −10.31484882945835729920666228854, −9.171587232911897801404638997315, −8.368959940213086626643115476541, −7.07962059422872321217136881094, −6.42699921247169900254366987571, −4.65888853970064614124319613983, −2.90614385001189915940597305988, −0.825231437707361500689070719825,
1.09279871682301673365817859951, 2.48639619216479626647330783997, 4.64057858316050085799363929560, 5.82869234818299649542765357826, 7.30623722906683857936290813055, 8.577012281878518954996358697780, 9.367870407755507033704941304808, 9.898693764368672213599421650447, 11.19195592676506662047462883969, 11.97432941394096392156063952602
