| L(s) = 1 | + (0.965 + 0.258i)2-s + (0.608 + 0.793i)3-s + (0.866 + 0.499i)4-s + (−0.309 − 2.34i)5-s + (0.382 + 0.923i)6-s + (2.57 + 0.622i)7-s + (0.707 + 0.707i)8-s + (−0.258 + 0.965i)9-s + (0.309 − 2.34i)10-s + (−1.37 − 0.181i)11-s + (0.130 + 0.991i)12-s + 0.134i·13-s + (2.32 + 1.26i)14-s + (1.67 − 1.67i)15-s + (0.500 + 0.866i)16-s + (4.07 − 0.609i)17-s + ⋯ |
| L(s) = 1 | + (0.683 + 0.183i)2-s + (0.351 + 0.458i)3-s + (0.433 + 0.249i)4-s + (−0.138 − 1.04i)5-s + (0.156 + 0.377i)6-s + (0.971 + 0.235i)7-s + (0.249 + 0.249i)8-s + (−0.0862 + 0.321i)9-s + (0.0977 − 0.742i)10-s + (−0.415 − 0.0547i)11-s + (0.0376 + 0.286i)12-s + 0.0372i·13-s + (0.620 + 0.338i)14-s + (0.432 − 0.432i)15-s + (0.125 + 0.216i)16-s + (0.989 − 0.147i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.71936 + 0.433279i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.71936 + 0.433279i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.965 - 0.258i)T \) |
| 3 | \( 1 + (-0.608 - 0.793i)T \) |
| 7 | \( 1 + (-2.57 - 0.622i)T \) |
| 17 | \( 1 + (-4.07 + 0.609i)T \) |
| good | 5 | \( 1 + (0.309 + 2.34i)T + (-4.82 + 1.29i)T^{2} \) |
| 11 | \( 1 + (1.37 + 0.181i)T + (10.6 + 2.84i)T^{2} \) |
| 13 | \( 1 - 0.134iT - 13T^{2} \) |
| 19 | \( 1 + (-3.87 - 1.03i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-1.14 + 1.48i)T + (-5.95 - 22.2i)T^{2} \) |
| 29 | \( 1 + (6.29 + 2.60i)T + (20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + (-4.41 - 5.75i)T + (-8.02 + 29.9i)T^{2} \) |
| 37 | \( 1 + (-0.0607 + 0.00799i)T + (35.7 - 9.57i)T^{2} \) |
| 41 | \( 1 + (9.51 - 3.93i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (8.37 + 8.37i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.16 - 0.671i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.204 - 0.761i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-7.37 + 1.97i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (2.15 + 1.65i)T + (15.7 + 58.9i)T^{2} \) |
| 67 | \( 1 + (-0.382 + 0.662i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.92 + 7.06i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (8.45 - 6.48i)T + (18.8 - 70.5i)T^{2} \) |
| 79 | \( 1 + (8.43 - 10.9i)T + (-20.4 - 76.3i)T^{2} \) |
| 83 | \( 1 + (5.69 - 5.69i)T - 83iT^{2} \) |
| 89 | \( 1 + (11.1 - 6.41i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.76 + 3.21i)T + (68.5 + 68.5i)T^{2} \) |
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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
−10.47161976438480064782468128564, −9.580834906299464775027971765672, −8.456232795874823993279725364327, −8.113962158606888871265033143609, −7.02587312973229735772454409217, −5.38921965979889950901549343737, −5.18672423182285601977883611251, −4.15281591288989176942956490756, −3.01835593955147370857816531531, −1.50991662388518879295614601550,
1.53230350262132519972351889072, 2.81353182493328873948147259302, 3.61751373069472641372437617509, 4.93642881801538350745160685671, 5.84250107639203034537967615696, 7.04884692697520690209912966992, 7.52209835877524385331832749569, 8.401955010518108294692374810763, 9.783164826803190984872296267806, 10.51333082638341563492230524570
