| L(s) = 1 | + (1.58 + 2.17i)2-s + (−12.6 − 38.8i)3-s + (17.5 − 53.9i)4-s + (114. + 83.3i)5-s + (64.6 − 88.9i)6-s + (−318. − 103. i)7-s + (308. − 100. i)8-s + (−761. + 553. i)9-s + 381. i·10-s + (1.27e3 + 374. i)11-s − 2.32e3·12-s + (1.89e3 + 2.60e3i)13-s + (−278. − 856. i)14-s + (1.79e3 − 5.51e3i)15-s + (−2.23e3 − 1.62e3i)16-s + (−410. + 565. i)17-s + ⋯ |
| L(s) = 1 | + (0.197 + 0.272i)2-s + (−0.467 − 1.43i)3-s + (0.274 − 0.843i)4-s + (0.917 + 0.666i)5-s + (0.299 − 0.411i)6-s + (−0.928 − 0.301i)7-s + (0.603 − 0.196i)8-s + (−1.04 + 0.759i)9-s + 0.381i·10-s + (0.959 + 0.281i)11-s − 1.34·12-s + (0.860 + 1.18i)13-s + (−0.101 − 0.312i)14-s + (0.530 − 1.63i)15-s + (−0.544 − 0.395i)16-s + (−0.0836 + 0.115i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.20309 - 0.786902i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20309 - 0.786902i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-1.27e3 - 374. i)T \) |
| good | 2 | \( 1 + (-1.58 - 2.17i)T + (-19.7 + 60.8i)T^{2} \) |
| 3 | \( 1 + (12.6 + 38.8i)T + (-589. + 428. i)T^{2} \) |
| 5 | \( 1 + (-114. - 83.3i)T + (4.82e3 + 1.48e4i)T^{2} \) |
| 7 | \( 1 + (318. + 103. i)T + (9.51e4 + 6.91e4i)T^{2} \) |
| 13 | \( 1 + (-1.89e3 - 2.60e3i)T + (-1.49e6 + 4.59e6i)T^{2} \) |
| 17 | \( 1 + (410. - 565. i)T + (-7.45e6 - 2.29e7i)T^{2} \) |
| 19 | \( 1 + (-5.50e3 + 1.78e3i)T + (3.80e7 - 2.76e7i)T^{2} \) |
| 23 | \( 1 + 1.41e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + (-1.70e4 - 5.53e3i)T + (4.81e8 + 3.49e8i)T^{2} \) |
| 31 | \( 1 + (6.11e3 - 4.44e3i)T + (2.74e8 - 8.44e8i)T^{2} \) |
| 37 | \( 1 + (1.61e4 - 4.97e4i)T + (-2.07e9 - 1.50e9i)T^{2} \) |
| 41 | \( 1 + (6.28e4 - 2.04e4i)T + (3.84e9 - 2.79e9i)T^{2} \) |
| 43 | \( 1 - 1.34e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 + (4.32e3 + 1.33e4i)T + (-8.72e9 + 6.33e9i)T^{2} \) |
| 53 | \( 1 + (-7.64e4 + 5.55e4i)T + (6.84e9 - 2.10e10i)T^{2} \) |
| 59 | \( 1 + (-6.33e4 + 1.94e5i)T + (-3.41e10 - 2.47e10i)T^{2} \) |
| 61 | \( 1 + (-9.11e4 + 1.25e5i)T + (-1.59e10 - 4.89e10i)T^{2} \) |
| 67 | \( 1 + 3.27e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + (-3.30e5 - 2.40e5i)T + (3.95e10 + 1.21e11i)T^{2} \) |
| 73 | \( 1 + (5.81e5 + 1.89e5i)T + (1.22e11 + 8.89e10i)T^{2} \) |
| 79 | \( 1 + (-2.14e5 - 2.94e5i)T + (-7.51e10 + 2.31e11i)T^{2} \) |
| 83 | \( 1 + (-2.11e5 + 2.90e5i)T + (-1.01e11 - 3.10e11i)T^{2} \) |
| 89 | \( 1 - 3.43e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + (-7.28e4 + 5.29e4i)T + (2.57e11 - 7.92e11i)T^{2} \) |
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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
−18.84951734831095309276872874491, −17.86811048949865016667648924929, −16.31144005181700905520179095649, −14.20554557626306254283075615080, −13.45828570746195401411255952731, −11.63025239497362235227004298169, −9.851236493134296164909181993485, −6.70893427966570948421095245708, −6.31461479289339662565143303154, −1.56942824625953306392681864037,
3.65278600267412362253665425469, 5.74723505055731345106559525751, 8.983190158044388386943287002044, 10.32196622474096315042450022250, 12.02193262706996066243860702306, 13.51207518899607488146305627169, 15.76549157900034880041379202323, 16.51550308376833888101582456882, 17.60661399152884715914211341606, 20.13402244314266874643686004192
