| L(s) = 1 | + (0.913 + 0.406i)2-s + (0.0920 − 0.0668i)3-s + (0.669 + 0.743i)4-s + (0.825 − 0.175i)5-s + (0.111 − 0.0236i)6-s + (0.228 − 2.17i)7-s + (0.309 + 0.951i)8-s + (−0.923 + 2.84i)9-s + (0.825 + 0.175i)10-s − 2.33·11-s + (0.111 + 0.0236i)12-s + (0.690 + 1.19i)13-s + (1.09 − 1.88i)14-s + (0.0642 − 0.0713i)15-s + (−0.104 + 0.994i)16-s + (−3.60 − 3.99i)17-s + ⋯ |
| L(s) = 1 | + (0.645 + 0.287i)2-s + (0.0531 − 0.0386i)3-s + (0.334 + 0.371i)4-s + (0.369 − 0.0784i)5-s + (0.0454 − 0.00965i)6-s + (0.0862 − 0.820i)7-s + (0.109 + 0.336i)8-s + (−0.307 + 0.946i)9-s + (0.261 + 0.0554i)10-s − 0.703·11-s + (0.0321 + 0.00682i)12-s + (0.191 + 0.331i)13-s + (0.291 − 0.505i)14-s + (0.0165 − 0.0184i)15-s + (−0.0261 + 0.248i)16-s + (−0.873 − 0.970i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.47442 + 0.252435i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.47442 + 0.252435i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.913 - 0.406i)T \) |
| 61 | \( 1 + (-7.32 - 2.70i)T \) |
| good | 3 | \( 1 + (-0.0920 + 0.0668i)T + (0.927 - 2.85i)T^{2} \) |
| 5 | \( 1 + (-0.825 + 0.175i)T + (4.56 - 2.03i)T^{2} \) |
| 7 | \( 1 + (-0.228 + 2.17i)T + (-6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 + 2.33T + 11T^{2} \) |
| 13 | \( 1 + (-0.690 - 1.19i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.60 + 3.99i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (0.392 + 3.73i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.381 + 1.17i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (2.06 - 3.57i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.911 + 0.405i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-0.341 - 0.248i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-6.54 - 4.75i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-7.60 + 8.44i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (3.88 - 6.72i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.15 - 3.55i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (3.19 + 1.42i)T + (39.4 + 43.8i)T^{2} \) |
| 67 | \( 1 + (-6.65 + 1.41i)T + (61.2 - 27.2i)T^{2} \) |
| 71 | \( 1 + (8.79 + 1.87i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (-11.7 - 2.50i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (1.75 - 1.94i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-8.34 - 3.71i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (6.14 - 4.46i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.02 + 1.79i)T + (64.9 - 72.0i)T^{2} \) |
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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
−13.61754247710101831907161546864, −12.87876597220228863492329195256, −11.33803663496537951701438105015, −10.68489131101757969528576201683, −9.218002906350583508446782338775, −7.84705043772465481767687139495, −6.91836102269074151536488619179, −5.44451306960574386228045555531, −4.38892493557885416345648562987, −2.51839013336924002914880791535,
2.33660704094914397196719779994, 3.88145733736662417576297468884, 5.58169100600650609700778309495, 6.30350979606088298862573038793, 8.074399390438266880642517331715, 9.279250791710872279085578533993, 10.39017569813915411850509017200, 11.50504086576739497901711018150, 12.46807320881744435858327089894, 13.24512936069554152327856315681
