| L(s) = 1 | + (1.19 + 0.766i)2-s + (−0.142 − 0.989i)3-s + (0.00459 + 0.0100i)4-s + (0.815 + 0.239i)5-s + (0.589 − 1.28i)6-s + (−3.31 + 3.82i)7-s + (0.401 − 2.79i)8-s + (−0.959 + 0.281i)9-s + (0.788 + 0.910i)10-s + (−1.24 + 0.799i)11-s + (0.00929 − 0.00597i)12-s + (−0.666 − 0.769i)13-s + (−6.88 + 2.02i)14-s + (0.120 − 0.840i)15-s + (2.63 − 3.03i)16-s + (2.37 − 5.20i)17-s + ⋯ |
| L(s) = 1 | + (0.843 + 0.542i)2-s + (−0.0821 − 0.571i)3-s + (0.00229 + 0.00502i)4-s + (0.364 + 0.107i)5-s + (0.240 − 0.526i)6-s + (−1.25 + 1.44i)7-s + (0.141 − 0.987i)8-s + (−0.319 + 0.0939i)9-s + (0.249 + 0.287i)10-s + (−0.374 + 0.240i)11-s + (0.00268 − 0.00172i)12-s + (−0.184 − 0.213i)13-s + (−1.84 + 0.540i)14-s + (0.0312 − 0.217i)15-s + (0.658 − 0.759i)16-s + (0.576 − 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.197i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.19493 + 0.119011i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19493 + 0.119011i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.142 + 0.989i)T \) |
| 23 | \( 1 + (-4.33 + 2.05i)T \) |
| good | 2 | \( 1 + (-1.19 - 0.766i)T + (0.830 + 1.81i)T^{2} \) |
| 5 | \( 1 + (-0.815 - 0.239i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (3.31 - 3.82i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (1.24 - 0.799i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (0.666 + 0.769i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.37 + 5.20i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.89 - 4.15i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (1.19 - 2.61i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.751 - 5.22i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-0.443 + 0.130i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-6.61 - 1.94i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (0.690 + 4.80i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 4.11T + 47T^{2} \) |
| 53 | \( 1 + (1.84 - 2.13i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (1.38 + 1.60i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (1.39 - 9.67i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (4.61 + 2.96i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (10.6 + 6.84i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-0.730 - 1.60i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-1.53 - 1.76i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (15.7 - 4.63i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.44 - 10.0i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-10.8 - 3.17i)T + (81.6 + 52.4i)T^{2} \) |
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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
−14.69806947355072710116503828500, −13.70950701746407064333704760918, −12.69291097349735724606204109150, −12.11280897132163564320777923326, −10.09798900939430700800718847919, −9.127339515151837173626661507506, −7.25537149169146121503474680854, −6.07259750272632187537929026829, −5.31220395210038793460464043484, −2.95234937670013805641573955137,
3.22966776821655297302951739269, 4.28892501015580432760587960297, 5.86286682214431987800455141849, 7.55973880319753180081840846116, 9.347070103781286120044567799847, 10.39432428064739708436428801794, 11.37618856239888511039696257497, 12.94951456337755605139430394853, 13.32133187928286200252511899148, 14.39163012810204263751681429367
