| L(s) = 1 | + (−0.463 − 1.33i)2-s + (2.94 + 1.37i)3-s + (−1.57 + 1.23i)4-s + (−1.38 − 1.97i)5-s + (0.469 − 4.56i)6-s + (1.46 + 2.54i)7-s + (2.38 + 1.52i)8-s + (4.85 + 5.78i)9-s + (−1.99 + 2.76i)10-s + (3.39 − 0.908i)11-s + (−6.32 + 1.48i)12-s + (−2.05 + 0.958i)13-s + (2.72 − 3.14i)14-s + (−1.35 − 7.70i)15-s + (0.932 − 3.88i)16-s + (−5.21 − 4.37i)17-s + ⋯ |
| L(s) = 1 | + (−0.327 − 0.944i)2-s + (1.69 + 0.792i)3-s + (−0.785 + 0.619i)4-s + (−0.617 − 0.882i)5-s + (0.191 − 1.86i)6-s + (0.555 + 0.962i)7-s + (0.842 + 0.538i)8-s + (1.61 + 1.92i)9-s + (−0.631 + 0.872i)10-s + (1.02 − 0.273i)11-s + (−1.82 + 0.430i)12-s + (−0.570 + 0.265i)13-s + (0.727 − 0.840i)14-s + (−0.350 − 1.98i)15-s + (0.233 − 0.972i)16-s + (−1.26 − 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.68419 - 0.290425i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.68419 - 0.290425i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.463 + 1.33i)T \) |
| 19 | \( 1 + (-3.40 - 2.72i)T \) |
| good | 3 | \( 1 + (-2.94 - 1.37i)T + (1.92 + 2.29i)T^{2} \) |
| 5 | \( 1 + (1.38 + 1.97i)T + (-1.71 + 4.69i)T^{2} \) |
| 7 | \( 1 + (-1.46 - 2.54i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.39 + 0.908i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (2.05 - 0.958i)T + (8.35 - 9.95i)T^{2} \) |
| 17 | \( 1 + (5.21 + 4.37i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (0.997 + 5.65i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.213 - 2.43i)T + (-28.5 - 5.03i)T^{2} \) |
| 31 | \( 1 + (-0.381 - 0.661i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.50 + 5.50i)T + 37iT^{2} \) |
| 41 | \( 1 + (0.138 - 0.0502i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (4.17 - 2.92i)T + (14.7 - 40.4i)T^{2} \) |
| 47 | \( 1 + (-2.52 - 3.01i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.472 + 0.675i)T + (-18.1 - 49.8i)T^{2} \) |
| 59 | \( 1 + (8.11 - 0.710i)T + (58.1 - 10.2i)T^{2} \) |
| 61 | \( 1 + (2.95 - 4.22i)T + (-20.8 - 57.3i)T^{2} \) |
| 67 | \( 1 + (12.3 + 1.07i)T + (65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (9.87 + 1.74i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (1.98 + 5.44i)T + (-55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-7.90 + 2.87i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (0.926 + 0.248i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-9.66 - 3.51i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-1.67 + 1.99i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−11.78850931071519233260377539538, −10.59257726090030918558280889946, −9.378903870183335106608453565242, −8.945728898616914584442736008959, −8.470199693638137294787344106014, −7.47002079707337756772854655868, −4.84918593227831041156912986274, −4.29325508629721943112892339427, −3.06177654059522920910690248585, −1.91881598724376150822196301888,
1.60451703601359926575799698085, 3.45332482029757925347508322919, 4.35294699170721569905752243599, 6.56089409263503559023896482506, 7.24931278941357128286030031544, 7.69295540916651753921684475672, 8.660037448468439566213312498374, 9.505239715216091815437531567234, 10.55253229383159993872179573115, 11.84628974774419693298567422500
