| L(s) = 1 | + (−0.697 − 0.335i)2-s + (0.634 + 1.61i)3-s + (−0.873 − 1.09i)4-s + (−2.42 + 2.24i)5-s + (0.0983 − 1.33i)6-s + (3.51 + 2.03i)7-s + (0.585 + 2.56i)8-s + (−2.19 + 2.04i)9-s + (2.44 − 0.752i)10-s + (1.58 + 1.26i)11-s + (1.21 − 2.10i)12-s + (−4.52 − 1.39i)13-s + (−1.76 − 2.59i)14-s + (−5.15 − 2.47i)15-s + (−0.170 + 0.747i)16-s + (1.47 − 1.58i)17-s + ⋯ |
| L(s) = 1 | + (−0.492 − 0.237i)2-s + (0.366 + 0.930i)3-s + (−0.436 − 0.547i)4-s + (−1.08 + 1.00i)5-s + (0.0401 − 0.545i)6-s + (1.32 + 0.767i)7-s + (0.207 + 0.907i)8-s + (−0.731 + 0.682i)9-s + (0.771 − 0.238i)10-s + (0.478 + 0.381i)11-s + (0.349 − 0.607i)12-s + (−1.25 − 0.386i)13-s + (−0.472 − 0.693i)14-s + (−1.33 − 0.638i)15-s + (−0.0426 + 0.186i)16-s + (0.356 − 0.384i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.167 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.580638 + 0.490464i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.580638 + 0.490464i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.634 - 1.61i)T \) |
| 43 | \( 1 + (2.12 - 6.20i)T \) |
| good | 2 | \( 1 + (0.697 + 0.335i)T + (1.24 + 1.56i)T^{2} \) |
| 5 | \( 1 + (2.42 - 2.24i)T + (0.373 - 4.98i)T^{2} \) |
| 7 | \( 1 + (-3.51 - 2.03i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.58 - 1.26i)T + (2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (4.52 + 1.39i)T + (10.7 + 7.32i)T^{2} \) |
| 17 | \( 1 + (-1.47 + 1.58i)T + (-1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-5.06 - 1.98i)T + (13.9 + 12.9i)T^{2} \) |
| 23 | \( 1 + (-1.02 + 6.77i)T + (-21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (0.531 - 0.362i)T + (10.5 - 26.9i)T^{2} \) |
| 31 | \( 1 + (-0.106 - 1.42i)T + (-30.6 + 4.62i)T^{2} \) |
| 37 | \( 1 + (-5.94 + 3.43i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.589 + 1.22i)T + (-25.5 - 32.0i)T^{2} \) |
| 47 | \( 1 + (-2.74 + 2.18i)T + (10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (-0.842 - 2.73i)T + (-43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-9.41 - 2.14i)T + (53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (-0.681 - 0.0510i)T + (60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (-0.172 + 0.440i)T + (-49.1 - 45.5i)T^{2} \) |
| 71 | \( 1 + (7.95 - 1.19i)T + (67.8 - 20.9i)T^{2} \) |
| 73 | \( 1 + (1.86 - 6.04i)T + (-60.3 - 41.1i)T^{2} \) |
| 79 | \( 1 + (-4.55 + 7.88i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.96 - 4.34i)T + (-30.3 - 77.2i)T^{2} \) |
| 89 | \( 1 + (-9.67 - 6.59i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (-2.46 + 3.09i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−14.20824923573773714510804484114, −12.01316846087333220012304788235, −11.31310829641972434996067320111, −10.42546301914347159007874378290, −9.490165008542287539155421953229, −8.359916362488938716207276833877, −7.53287956623411819907735591534, −5.35886865540523039157323168221, −4.40426409731120657380793654091, −2.62691979180988356752261450364,
1.02084804499831699621126199192, 3.75619885050094244349364711991, 4.94807252311456638240919954485, 7.32818180199814597407948968705, 7.69155056546303128688006890414, 8.530984463468023099202093949375, 9.502137714843320649171564366440, 11.58480173075327072762880442583, 11.93865005890375371688303670309, 13.11247335306554446946908980675
