| L(s) = 1 | + (−0.118 − 0.0570i)2-s + (−1.23 − 1.55i)4-s + (1.30 − 1.21i)5-s + (−0.00749 + 0.0129i)7-s + (0.116 + 0.510i)8-s + (−0.223 + 0.0689i)10-s + (1.29 − 1.62i)11-s + (−1.55 − 0.478i)13-s + (0.00162 − 0.00110i)14-s + (−0.867 + 3.79i)16-s + (−4.42 − 4.10i)17-s + (2.78 − 7.09i)19-s + (−3.49 − 0.526i)20-s + (−0.246 + 0.118i)22-s + (−5.24 − 0.790i)23-s + ⋯ |
| L(s) = 1 | + (−0.0837 − 0.0403i)2-s + (−0.618 − 0.775i)4-s + (0.583 − 0.541i)5-s + (−0.00283 + 0.00490i)7-s + (0.0411 + 0.180i)8-s + (−0.0707 + 0.0218i)10-s + (0.390 − 0.490i)11-s + (−0.430 − 0.132i)13-s + (0.000434 − 0.000296i)14-s + (−0.216 + 0.949i)16-s + (−1.07 − 0.995i)17-s + (0.638 − 1.62i)19-s + (−0.780 − 0.117i)20-s + (−0.0524 + 0.0252i)22-s + (−1.09 − 0.164i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.244 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.244 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.667052 - 0.856459i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.667052 - 0.856459i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + (1.84 + 6.29i)T \) |
| good | 2 | \( 1 + (0.118 + 0.0570i)T + (1.24 + 1.56i)T^{2} \) |
| 5 | \( 1 + (-1.30 + 1.21i)T + (0.373 - 4.98i)T^{2} \) |
| 7 | \( 1 + (0.00749 - 0.0129i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.29 + 1.62i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (1.55 + 0.478i)T + (10.7 + 7.32i)T^{2} \) |
| 17 | \( 1 + (4.42 + 4.10i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-2.78 + 7.09i)T + (-13.9 - 12.9i)T^{2} \) |
| 23 | \( 1 + (5.24 + 0.790i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-5.92 + 4.04i)T + (10.5 - 26.9i)T^{2} \) |
| 31 | \( 1 + (-0.178 - 2.37i)T + (-30.6 + 4.62i)T^{2} \) |
| 37 | \( 1 + (2.52 + 4.36i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.20 - 1.54i)T + (25.5 + 32.0i)T^{2} \) |
| 47 | \( 1 + (-6.24 - 7.83i)T + (-10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (-6.55 + 2.02i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (0.370 - 1.62i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (0.452 - 6.03i)T + (-60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (-0.523 + 1.33i)T + (-49.1 - 45.5i)T^{2} \) |
| 71 | \( 1 + (6.96 - 1.05i)T + (67.8 - 20.9i)T^{2} \) |
| 73 | \( 1 + (-9.32 - 2.87i)T + (60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 + (6.00 - 10.4i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7.49 + 5.10i)T + (30.3 + 77.2i)T^{2} \) |
| 89 | \( 1 + (-13.1 - 8.95i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (-8.44 + 10.5i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−10.98218787919475707149797817534, −9.988134220720882586147189694860, −9.171563657675598103075690321163, −8.739060309706557441563972916786, −7.22912241656963987297444106672, −6.09343928780213708372850053226, −5.16764123316402681446306362009, −4.34378397583824542129111870415, −2.45197980387132350420854911629, −0.76778262201683458932151272073,
2.08725010117758566675685219771, 3.56852906894338069644791838151, 4.53788907190771663509804023756, 5.94362913557912703843395680571, 6.91290163820583335615992783241, 7.962862450187465799535084520345, 8.786074905933085376472289343981, 9.880614620054302189264872579258, 10.37426989173056206720185572321, 11.85339038511706646762566496144
