| L(s) = 1 | + (0.161 − 1.53i)2-s + (0.550 + 1.64i)3-s + (−0.379 − 0.0806i)4-s − 1.41·5-s + (2.61 − 0.581i)6-s + (1.37 + 4.22i)7-s + (0.769 − 2.36i)8-s + (−2.39 + 1.80i)9-s + (−0.228 + 2.17i)10-s + (4.07 + 0.866i)11-s + (−0.0765 − 0.667i)12-s + (−1.26 + 0.918i)13-s + (6.70 − 1.42i)14-s + (−0.778 − 2.31i)15-s + (−4.22 − 1.88i)16-s + (−1.19 − 1.32i)17-s + ⋯ |
| L(s) = 1 | + (0.114 − 1.08i)2-s + (0.318 + 0.948i)3-s + (−0.189 − 0.0403i)4-s − 0.631·5-s + (1.06 − 0.237i)6-s + (0.518 + 1.59i)7-s + (0.272 − 0.837i)8-s + (−0.797 + 0.603i)9-s + (−0.0721 + 0.686i)10-s + (1.22 + 0.261i)11-s + (−0.0221 − 0.192i)12-s + (−0.350 + 0.254i)13-s + (1.79 − 0.381i)14-s + (−0.200 − 0.598i)15-s + (−1.05 − 0.470i)16-s + (−0.289 − 0.321i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0395i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.55635 + 0.0308044i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.55635 + 0.0308044i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.550 - 1.64i)T \) |
| 31 | \( 1 + (-3.52 - 4.31i)T \) |
| good | 2 | \( 1 + (-0.161 + 1.53i)T + (-1.95 - 0.415i)T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 + (-1.37 - 4.22i)T + (-5.66 + 4.11i)T^{2} \) |
| 11 | \( 1 + (-4.07 - 0.866i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (1.26 - 0.918i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (1.19 + 1.32i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-5.06 + 2.25i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-1.76 - 1.95i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.456 + 4.33i)T + (-28.3 - 6.02i)T^{2} \) |
| 37 | \( 1 + (-4.15 + 7.20i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (6.39 + 4.64i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (9.48 + 6.89i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (3.40 + 1.51i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (2.83 - 0.602i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-3.78 - 1.68i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (-5.16 - 8.95i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 - 6.35T + 67T^{2} \) |
| 71 | \( 1 + (-8.09 + 1.72i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (2.56 - 2.85i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (0.459 - 1.41i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (13.9 - 6.19i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (2.59 + 7.98i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-7.24 + 8.05i)T + (-10.1 - 96.4i)T^{2} \) |
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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
−11.58732040375901139736205279689, −11.41849968731007172909184507825, −9.929182902573761045235609061456, −9.285116526938385325675674248195, −8.445750943881886251441829535586, −7.03448047922464063098133611645, −5.41143990330196292714815015551, −4.31687001837416050427476753025, −3.24252105986925926670057256038, −2.10288935473942946418236319519,
1.33031002417804878009653792339, 3.50417913364258804997891251427, 4.80694421395323266031846170143, 6.37808045823482568655363639569, 6.95796052837210976128768153834, 7.87338246099879114486716793590, 8.274053622377084552680186490291, 9.842626344166156011188722597382, 11.34029871737426794479773045436, 11.65134888721242752643654507201
