| L(s) = 1 | + (2.57 − 0.103i)2-s + (−0.381 − 1.31i)3-s + (4.64 − 0.374i)4-s + (−3.32 + 1.26i)5-s + (−1.12 − 3.35i)6-s + (0.633 − 1.48i)7-s + (6.81 − 0.827i)8-s + (0.946 − 0.598i)9-s + (−8.44 + 3.59i)10-s + (−2.92 + 4.61i)11-s + (−2.26 − 5.97i)12-s + (0.0292 + 3.60i)13-s + (1.47 − 3.90i)14-s + (2.92 + 3.89i)15-s + (8.28 − 1.34i)16-s + (0.607 + 0.258i)17-s + ⋯ |
| L(s) = 1 | + (1.82 − 0.0734i)2-s + (−0.220 − 0.760i)3-s + (2.32 − 0.187i)4-s + (−1.48 + 0.563i)5-s + (−0.457 − 1.37i)6-s + (0.239 − 0.562i)7-s + (2.40 − 0.292i)8-s + (0.315 − 0.199i)9-s + (−2.66 + 1.13i)10-s + (−0.880 + 1.39i)11-s + (−0.654 − 1.72i)12-s + (0.00810 + 0.999i)13-s + (0.395 − 1.04i)14-s + (0.756 + 1.00i)15-s + (2.07 − 0.336i)16-s + (0.147 + 0.0628i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.29606 - 0.614350i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.29606 - 0.614350i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-0.0292 - 3.60i)T \) |
| good | 2 | \( 1 + (-2.57 + 0.103i)T + (1.99 - 0.160i)T^{2} \) |
| 3 | \( 1 + (0.381 + 1.31i)T + (-2.53 + 1.60i)T^{2} \) |
| 5 | \( 1 + (3.32 - 1.26i)T + (3.74 - 3.31i)T^{2} \) |
| 7 | \( 1 + (-0.633 + 1.48i)T + (-4.84 - 5.04i)T^{2} \) |
| 11 | \( 1 + (2.92 - 4.61i)T + (-4.71 - 9.93i)T^{2} \) |
| 17 | \( 1 + (-0.607 - 0.258i)T + (11.7 + 12.2i)T^{2} \) |
| 19 | \( 1 + (3.26 + 1.88i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.06 + 3.58i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.0150 + 0.374i)T + (-28.9 + 2.33i)T^{2} \) |
| 31 | \( 1 + (-4.01 + 4.52i)T + (-3.73 - 30.7i)T^{2} \) |
| 37 | \( 1 + (-8.68 - 1.77i)T + (34.0 + 14.5i)T^{2} \) |
| 41 | \( 1 + (6.39 - 1.85i)T + (34.6 - 21.9i)T^{2} \) |
| 43 | \( 1 + (1.58 + 7.75i)T + (-39.5 + 16.8i)T^{2} \) |
| 47 | \( 1 + (-7.99 - 5.51i)T + (16.6 + 43.9i)T^{2} \) |
| 53 | \( 1 + (0.387 + 3.18i)T + (-51.4 + 12.6i)T^{2} \) |
| 59 | \( 1 + (-0.599 + 3.68i)T + (-55.9 - 18.6i)T^{2} \) |
| 61 | \( 1 + (3.97 + 2.98i)T + (16.9 + 58.5i)T^{2} \) |
| 67 | \( 1 + (0.700 - 8.67i)T + (-66.1 - 10.7i)T^{2} \) |
| 71 | \( 1 + (-0.998 - 0.959i)T + (2.85 + 70.9i)T^{2} \) |
| 73 | \( 1 + (0.592 + 1.12i)T + (-41.4 + 60.0i)T^{2} \) |
| 79 | \( 1 + (2.22 - 3.22i)T + (-28.0 - 73.8i)T^{2} \) |
| 83 | \( 1 + (1.29 + 5.24i)T + (-73.4 + 38.5i)T^{2} \) |
| 89 | \( 1 + (-2.98 + 1.72i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (10.3 - 8.45i)T + (19.4 - 95.0i)T^{2} \) |
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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
−12.64640265602941064829560031833, −12.01635330059612169693068059651, −11.31750203883927704914908192749, −10.29030827759539380149917148403, −7.79874151829973167106590401954, −7.15109176609073055423611246800, −6.43600089209193469749301192812, −4.54933281947871801293271547763, −4.07672747787705185962972979794, −2.35291043309733468343661329194,
3.10954806284263771463929008899, 4.08277376286960215430961535525, 5.06005798576170379983312532180, 5.83006024328164707892476110919, 7.60024283378501980898845480484, 8.397000478131389877952744805818, 10.48701012263283796500286252108, 11.22847234957584780590050571639, 12.04054084106673080322715294566, 12.84547465457740212428683348226
