| L(s) = 1 | + (−0.662 − 1.45i)2-s + (0.959 − 0.281i)3-s + (−0.354 + 0.409i)4-s + (−0.438 − 0.281i)5-s + (−1.04 − 1.20i)6-s + (0.188 + 1.31i)7-s + (−2.23 − 0.654i)8-s + (0.841 − 0.540i)9-s + (−0.118 + 0.822i)10-s + (0.0950 − 0.208i)11-s + (−0.225 + 0.492i)12-s + (−0.382 + 2.65i)13-s + (1.77 − 1.14i)14-s + (−0.499 − 0.146i)15-s + (0.681 + 4.74i)16-s + (3.83 + 4.42i)17-s + ⋯ |
| L(s) = 1 | + (−0.468 − 1.02i)2-s + (0.553 − 0.162i)3-s + (−0.177 + 0.204i)4-s + (−0.196 − 0.125i)5-s + (−0.426 − 0.491i)6-s + (0.0712 + 0.495i)7-s + (−0.788 − 0.231i)8-s + (0.280 − 0.180i)9-s + (−0.0373 + 0.260i)10-s + (0.0286 − 0.0627i)11-s + (−0.0649 + 0.142i)12-s + (−0.105 + 0.737i)13-s + (0.474 − 0.305i)14-s + (−0.129 − 0.0379i)15-s + (0.170 + 1.18i)16-s + (0.929 + 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.104 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.631055 - 0.568320i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.631055 - 0.568320i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.959 + 0.281i)T \) |
| 23 | \( 1 + (0.745 - 4.73i)T \) |
| good | 2 | \( 1 + (0.662 + 1.45i)T + (-1.30 + 1.51i)T^{2} \) |
| 5 | \( 1 + (0.438 + 0.281i)T + (2.07 + 4.54i)T^{2} \) |
| 7 | \( 1 + (-0.188 - 1.31i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (-0.0950 + 0.208i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (0.382 - 2.65i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-3.83 - 4.42i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-1.30 + 1.50i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (4.19 + 4.83i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (9.52 + 2.79i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-7.40 + 4.75i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (-0.614 - 0.395i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (9.98 - 2.93i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + 2.04T + 47T^{2} \) |
| 53 | \( 1 + (-0.955 - 6.64i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-0.928 + 6.45i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-2.99 - 0.878i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (4.63 + 10.1i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (2.37 + 5.19i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (1.53 - 1.76i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (1.38 - 9.64i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (0.303 - 0.194i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-14.2 + 4.17i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-4.02 - 2.58i)T + (40.2 + 88.2i)T^{2} \) |
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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
−14.54973077526827837602998908782, −13.12516824486544597310543865051, −12.08824528588528792003448105077, −11.24259925055538131213121165961, −9.867629006915898589822998131981, −9.057002484274731951050268516142, −7.76902983712440252194838208888, −5.96729268497104026043874092327, −3.69176065791913204470937826730, −1.98356146236179602396328766504,
3.29938586733823977250338832436, 5.40046090275397100460822961372, 7.11234274359037003920351567863, 7.80654276939839532533405431171, 9.014074049172183334838385567946, 10.18492646617560665619407386951, 11.67863516610654393647019539926, 13.02953701738324537990697761884, 14.42889420384629081092665598150, 14.98128338535346255438114332810
