| L(s) = 1 | + (−0.217 + 1.51i)5-s + (−1.55 − 3.40i)7-s + (−1.04 + 0.306i)11-s + (1.25 − 2.75i)13-s + (−2.61 + 1.67i)17-s + (−3.49 − 2.24i)19-s + (−2.35 − 4.17i)23-s + (2.56 + 0.752i)25-s + (−2.24 + 1.44i)29-s + (−4.45 − 5.14i)31-s + (5.48 − 1.61i)35-s + (−0.973 − 6.77i)37-s + (1.41 − 9.85i)41-s + (−2.62 + 3.02i)43-s + 1.56·47-s + ⋯ |
| L(s) = 1 | + (−0.0971 + 0.675i)5-s + (−0.588 − 1.28i)7-s + (−0.314 + 0.0924i)11-s + (0.348 − 0.763i)13-s + (−0.633 + 0.406i)17-s + (−0.801 − 0.514i)19-s + (−0.490 − 0.871i)23-s + (0.512 + 0.150i)25-s + (−0.416 + 0.267i)29-s + (−0.800 − 0.923i)31-s + (0.927 − 0.272i)35-s + (−0.160 − 1.11i)37-s + (0.221 − 1.53i)41-s + (−0.400 + 0.461i)43-s + 0.228·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.417313 - 0.675244i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.417313 - 0.675244i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 + (2.35 + 4.17i)T \) |
| good | 5 | \( 1 + (0.217 - 1.51i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (1.55 + 3.40i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (1.04 - 0.306i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.25 + 2.75i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (2.61 - 1.67i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (3.49 + 2.24i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (2.24 - 1.44i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (4.45 + 5.14i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (0.973 + 6.77i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-1.41 + 9.85i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (2.62 - 3.02i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 1.56T + 47T^{2} \) |
| 53 | \( 1 + (-1.19 - 2.61i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-6.11 + 13.3i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-3.42 - 3.95i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-4.44 - 1.30i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (14.6 + 4.29i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-2.70 - 1.73i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (1.28 - 2.81i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.773 - 5.38i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (9.86 - 11.3i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.546 + 3.79i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−10.25181662618333560986101819084, −9.143707809753131340704244895839, −8.159365248712127985742597839818, −7.20386381160538021680049421761, −6.68390847273425022018878327375, −5.64640674221238487604139362845, −4.27148379673841998709980685893, −3.55310076163261547469144305014, −2.32391742539484829315665752180, −0.37288453192881885009173836626,
1.76134983115881343890609063648, 2.96408856027377609103452514726, 4.22399412074403138896395593508, 5.25234067594319645070775589910, 6.06742751451221084839241276833, 6.93846343798994477759047507040, 8.233971065606404055181415137943, 8.834722815779521025778337867870, 9.437871164932807709472460495844, 10.41123107878354117608301631377
