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
−11.97432941394096392156063952602, −11.19195592676506662047462883969, −9.898693764368672213599421650447, −9.367870407755507033704941304808, −8.577012281878518954996358697780, −7.30623722906683857936290813055, −5.82869234818299649542765357826, −4.64057858316050085799363929560, −2.48639619216479626647330783997, −1.09279871682301673365817859951,
0.825231437707361500689070719825, 2.90614385001189915940597305988, 4.65888853970064614124319613983, 6.42699921247169900254366987571, 7.07962059422872321217136881094, 8.368959940213086626643115476541, 9.171587232911897801404638997315, 10.31484882945835729920666228854, 10.73478530960684399982840068428, 12.38953280368022229977906688846