| L(s) = 1 | + (−0.427 + 1.87i)2-s + (−1.37 − 1.04i)3-s + (−1.51 − 0.730i)4-s + (−2.44 − 0.367i)5-s + (2.54 − 2.13i)6-s + (−3.43 − 1.98i)7-s + (−0.379 + 0.475i)8-s + (0.804 + 2.89i)9-s + (1.73 − 4.41i)10-s + (−1.56 − 3.24i)11-s + (1.32 + 2.59i)12-s + (1.33 + 3.40i)13-s + (5.17 − 5.57i)14-s + (2.98 + 3.06i)15-s + (−2.82 − 3.54i)16-s + (0.256 + 1.70i)17-s + ⋯ |
| L(s) = 1 | + (−0.301 + 1.32i)2-s + (−0.796 − 0.604i)3-s + (−0.758 − 0.365i)4-s + (−1.09 − 0.164i)5-s + (1.04 − 0.870i)6-s + (−1.29 − 0.748i)7-s + (−0.134 + 0.168i)8-s + (0.268 + 0.963i)9-s + (0.547 − 1.39i)10-s + (−0.470 − 0.977i)11-s + (0.382 + 0.749i)12-s + (0.370 + 0.944i)13-s + (1.38 − 1.48i)14-s + (0.769 + 0.791i)15-s + (−0.706 − 0.886i)16-s + (0.0622 + 0.412i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.497 + 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00239537 - 0.00413507i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00239537 - 0.00413507i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.37 + 1.04i)T \) |
| 43 | \( 1 + (2.98 - 5.83i)T \) |
| good | 2 | \( 1 + (0.427 - 1.87i)T + (-1.80 - 0.867i)T^{2} \) |
| 5 | \( 1 + (2.44 + 0.367i)T + (4.77 + 1.47i)T^{2} \) |
| 7 | \( 1 + (3.43 + 1.98i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.56 + 3.24i)T + (-6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (-1.33 - 3.40i)T + (-9.52 + 8.84i)T^{2} \) |
| 17 | \( 1 + (-0.256 - 1.70i)T + (-16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (-2.42 + 0.181i)T + (18.7 - 2.83i)T^{2} \) |
| 23 | \( 1 + (3.99 - 5.85i)T + (-8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (6.03 + 5.59i)T + (2.16 + 28.9i)T^{2} \) |
| 31 | \( 1 + (1.86 - 0.575i)T + (25.6 - 17.4i)T^{2} \) |
| 37 | \( 1 + (3.03 - 1.75i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.00 - 0.914i)T + (36.9 + 17.7i)T^{2} \) |
| 47 | \( 1 + (-3.99 + 8.30i)T + (-29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (2.21 + 0.867i)T + (38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-7.59 + 6.05i)T + (13.1 - 57.5i)T^{2} \) |
| 61 | \( 1 + (-0.492 + 1.59i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (0.927 + 12.3i)T + (-66.2 + 9.98i)T^{2} \) |
| 71 | \( 1 + (2.98 - 2.03i)T + (25.9 - 66.0i)T^{2} \) |
| 73 | \( 1 + (3.06 - 1.20i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (5.47 - 9.49i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.55 + 2.75i)T + (-6.20 + 82.7i)T^{2} \) |
| 89 | \( 1 + (6.53 - 6.06i)T + (6.65 - 88.7i)T^{2} \) |
| 97 | \( 1 + (3.00 - 1.44i)T + (60.4 - 75.8i)T^{2} \) |
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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
−13.22388246530403320771656313273, −11.88631903294677355084886066546, −11.15985135017986319851802377765, −9.645263861068240126822051355643, −8.184086032713154232978244131169, −7.44087674233336744576944878807, −6.53587714002261718220520556228, −5.61766126902600997297083619390, −3.80825201642575105545279789948, −0.00574329790105542524310710653,
2.96324134562103412899262738860, 4.02175172368305574930224633460, 5.72674159643377331829369636110, 7.16197697325049844335095765086, 8.926834941183279020477458457746, 9.913337599606113159584171338299, 10.59211347924602236015080532816, 11.61206937187885228560990327489, 12.42199405738670327522187852314, 12.81372004884170226629031840362
