| L(s) = 1 | + (0.0523 + 0.998i)2-s + (0.0281 − 0.0732i)3-s + (−0.994 + 0.104i)4-s + (0.0572 + 2.23i)5-s + (0.0745 + 0.0242i)6-s + (−2.64 − 0.147i)7-s + (−0.156 − 0.987i)8-s + (2.22 + 2.00i)9-s + (−2.22 + 0.174i)10-s + (−1.09 + 3.12i)11-s + (−0.0202 + 0.0757i)12-s + (−3.57 − 1.82i)13-s + (0.00867 − 2.64i)14-s + (0.165 + 0.0586i)15-s + (0.978 − 0.207i)16-s + (0.0428 − 0.817i)17-s + ⋯ |
| L(s) = 1 | + (0.0370 + 0.706i)2-s + (0.0162 − 0.0422i)3-s + (−0.497 + 0.0522i)4-s + (0.0256 + 0.999i)5-s + (0.0304 + 0.00989i)6-s + (−0.998 − 0.0556i)7-s + (−0.0553 − 0.349i)8-s + (0.741 + 0.667i)9-s + (−0.704 + 0.0550i)10-s + (−0.331 + 0.943i)11-s + (−0.00585 + 0.0218i)12-s + (−0.992 − 0.505i)13-s + (0.00231 − 0.707i)14-s + (0.0426 + 0.0151i)15-s + (0.244 − 0.0519i)16-s + (0.0103 − 0.198i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.145396 - 0.608824i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.145396 - 0.608824i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0523 - 0.998i)T \) |
| 5 | \( 1 + (-0.0572 - 2.23i)T \) |
| 7 | \( 1 + (2.64 + 0.147i)T \) |
| 11 | \( 1 + (1.09 - 3.12i)T \) |
| good | 3 | \( 1 + (-0.0281 + 0.0732i)T + (-2.22 - 2.00i)T^{2} \) |
| 13 | \( 1 + (3.57 + 1.82i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.0428 + 0.817i)T + (-16.9 - 1.77i)T^{2} \) |
| 19 | \( 1 + (-0.526 + 5.01i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (0.947 - 3.53i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (1.74 + 2.40i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (0.229 - 1.08i)T + (-28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (8.52 - 3.27i)T + (27.4 - 24.7i)T^{2} \) |
| 41 | \( 1 + (5.24 - 7.21i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (6.69 + 6.69i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.49 + 2.83i)T + (9.77 - 45.9i)T^{2} \) |
| 53 | \( 1 + (-5.04 - 7.76i)T + (-21.5 + 48.4i)T^{2} \) |
| 59 | \( 1 + (0.699 + 6.65i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (-1.46 - 6.88i)T + (-55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (1.27 + 4.76i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (3.12 - 9.61i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (1.15 + 0.935i)T + (15.1 + 71.4i)T^{2} \) |
| 79 | \( 1 + (3.10 + 2.79i)T + (8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (-1.53 - 3.01i)T + (-48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-5.03 - 8.71i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.579 + 1.13i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−10.28218803388874865156707672226, −10.13587975760368245640363343346, −9.235603169188405063861970696238, −7.88196377177073981104297396838, −7.05975996997233105797605975092, −6.89886338907689914296493191721, −5.53237617084072889550587000811, −4.65665588873352729757458324128, −3.38477372388742138155861351251, −2.29088019175577377103229905069,
0.29849905442693691411188631635, 1.80853136263953169977917232909, 3.31461693203186976093012771831, 4.10150659311855015825074978778, 5.21710871366684657997879352738, 6.14889391502467730897527008718, 7.25208992523510001019253939033, 8.457720147724949937684202644760, 9.108793890738109908334211961951, 9.915766357951867060038040966287
