| L(s) = 1 | + (0.237 + 1.50i)2-s + (0.710 − 0.361i)3-s + (−0.295 + 0.0959i)4-s + (−1.71 + 1.43i)5-s + (0.712 + 0.980i)6-s + (−0.0869 + 0.170i)7-s + (1.16 + 2.28i)8-s + (−1.38 + 1.91i)9-s + (−2.56 − 2.23i)10-s + (−0.175 + 0.175i)12-s + (−3.05 + 0.484i)13-s + (−0.276 − 0.0899i)14-s + (−0.698 + 1.63i)15-s + (−3.66 + 2.65i)16-s + (3.66 + 0.579i)17-s + (−3.20 − 1.63i)18-s + ⋯ |
| L(s) = 1 | + (0.168 + 1.06i)2-s + (0.410 − 0.208i)3-s + (−0.147 + 0.0479i)4-s + (−0.766 + 0.641i)5-s + (0.290 + 0.400i)6-s + (−0.0328 + 0.0644i)7-s + (0.412 + 0.808i)8-s + (−0.463 + 0.637i)9-s + (−0.810 − 0.706i)10-s + (−0.0505 + 0.0505i)12-s + (−0.847 + 0.134i)13-s + (−0.0739 − 0.0240i)14-s + (−0.180 + 0.423i)15-s + (−0.915 + 0.664i)16-s + (0.887 + 0.140i)17-s + (−0.754 − 0.384i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.445i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.340107 + 1.44821i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.340107 + 1.44821i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.71 - 1.43i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (-0.237 - 1.50i)T + (-1.90 + 0.618i)T^{2} \) |
| 3 | \( 1 + (-0.710 + 0.361i)T + (1.76 - 2.42i)T^{2} \) |
| 7 | \( 1 + (0.0869 - 0.170i)T + (-4.11 - 5.66i)T^{2} \) |
| 13 | \( 1 + (3.05 - 0.484i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-3.66 - 0.579i)T + (16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (0.229 - 0.707i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (1.14 + 1.14i)T + 23iT^{2} \) |
| 29 | \( 1 + (-2.95 - 9.07i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.283 + 0.206i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (4.81 + 2.45i)T + (21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (6.36 + 2.06i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-3.72 + 3.72i)T - 43iT^{2} \) |
| 47 | \( 1 + (-5.61 - 11.0i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (1.41 + 8.91i)T + (-50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (-9.15 + 2.97i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-3.46 - 4.76i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-4.13 + 4.13i)T - 67iT^{2} \) |
| 71 | \( 1 + (-9.27 + 6.73i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (2.14 + 1.09i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (0.542 + 0.394i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-2.60 + 16.4i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 - 7.92iT - 89T^{2} \) |
| 97 | \( 1 + (1.36 - 0.215i)T + (92.2 - 29.9i)T^{2} \) |
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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
−10.92980194341171883876874732900, −10.30620313688624064033758029251, −8.858099324976596167935250054684, −8.075676007527245964211062194541, −7.41846184483208732373210662023, −6.81295241921849497736517616163, −5.65514600558273172480617511428, −4.78229263709214861788845553720, −3.34442602746638038177417922570, −2.20575549441009387514692381124,
0.73222270224335200182207309227, 2.43224185768223646331651588062, 3.47582790839076146209382832011, 4.20054637347296186698490565328, 5.38363150416076641631639067765, 6.83201732142012724470969373055, 7.79592125790514057560913927586, 8.648514462425205617657412643964, 9.689965447843870727852932302809, 10.18787111642515371968734203913
