| L(s) = 1 | + (−1.90 + 0.559i)2-s + (−0.654 − 0.755i)3-s + (1.63 − 1.05i)4-s + (−0.291 − 2.02i)5-s + (1.67 + 1.07i)6-s + (1.34 − 2.93i)7-s + (0.0729 − 0.0841i)8-s + (−0.142 + 0.989i)9-s + (1.68 + 3.69i)10-s + (−1.55 − 0.457i)11-s + (−1.86 − 0.547i)12-s + (−1.07 − 2.36i)13-s + (−0.913 + 6.35i)14-s + (−1.34 + 1.54i)15-s + (−1.70 + 3.73i)16-s + (3.90 + 2.51i)17-s + ⋯ |
| L(s) = 1 | + (−1.34 + 0.395i)2-s + (−0.378 − 0.436i)3-s + (0.817 − 0.525i)4-s + (−0.130 − 0.906i)5-s + (0.682 + 0.438i)6-s + (0.507 − 1.11i)7-s + (0.0257 − 0.0297i)8-s + (−0.0474 + 0.329i)9-s + (0.534 + 1.16i)10-s + (−0.470 − 0.138i)11-s + (−0.538 − 0.158i)12-s + (−0.299 − 0.654i)13-s + (−0.244 + 1.69i)14-s + (−0.346 + 0.399i)15-s + (−0.426 + 0.934i)16-s + (0.947 + 0.609i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.347893 - 0.230519i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.347893 - 0.230519i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (-1.37 + 4.59i)T \) |
| good | 2 | \( 1 + (1.90 - 0.559i)T + (1.68 - 1.08i)T^{2} \) |
| 5 | \( 1 + (0.291 + 2.02i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.34 + 2.93i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (1.55 + 0.457i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (1.07 + 2.36i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-3.90 - 2.51i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (5.38 - 3.45i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-6.49 - 4.17i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-5.74 + 6.62i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (1.13 - 7.88i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.407 - 2.83i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-2.44 - 2.81i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 8.70T + 47T^{2} \) |
| 53 | \( 1 + (0.624 - 1.36i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-2.08 - 4.57i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-5.53 + 6.39i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (3.57 - 1.05i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-1.18 + 0.347i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-9.93 + 6.38i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-3.73 - 8.17i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-2.18 + 15.1i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-1.71 - 1.97i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (0.172 + 1.20i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−14.76116309769050783118395722745, −13.26958782836771138754821638018, −12.38012661456814581801319474241, −10.72860581389982202947207025983, −10.07965663567738312803750210005, −8.282867589842716708123630108581, −7.981209001871739310503674147900, −6.48737565548604064164791349203, −4.63021087335897201584213693447, −0.999569432276039913572734679715,
2.54658940710343168413876324229, 5.09555225416368616541562739900, 6.91215406863156894280474825556, 8.307434954579621001671614416480, 9.378027106399824765162242948955, 10.42312813955547236388287762093, 11.29812825593824744855413233662, 12.13789935081436155896635480402, 14.14420615146686200968938974169, 15.19060447335339126865108122801
