| L(s) = 1 | + (−0.296 − 0.142i)2-s + (−0.942 + 1.45i)3-s + (−1.17 − 1.47i)4-s + (−0.788 − 3.45i)5-s + (0.486 − 0.295i)6-s − 0.343i·7-s + (0.284 + 1.24i)8-s + (−1.22 − 2.73i)9-s + (−0.259 + 1.13i)10-s + (0.522 + 0.416i)11-s + (3.26 − 0.319i)12-s + (−1.40 − 6.15i)13-s + (−0.0490 + 0.101i)14-s + (5.76 + 2.11i)15-s + (−0.748 + 3.27i)16-s + (3.33 + 0.761i)17-s + ⋯ |
| L(s) = 1 | + (−0.209 − 0.100i)2-s + (−0.544 + 0.838i)3-s + (−0.589 − 0.739i)4-s + (−0.352 − 1.54i)5-s + (0.198 − 0.120i)6-s − 0.129i·7-s + (0.100 + 0.440i)8-s + (−0.407 − 0.913i)9-s + (−0.0819 + 0.358i)10-s + (0.157 + 0.125i)11-s + (0.941 − 0.0922i)12-s + (−0.389 − 1.70i)13-s + (−0.0131 + 0.0272i)14-s + (1.48 + 0.544i)15-s + (−0.187 + 0.819i)16-s + (0.808 + 0.184i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.352827 - 0.439071i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.352827 - 0.439071i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.942 - 1.45i)T \) |
| 43 | \( 1 + (-4.89 + 4.36i)T \) |
| good | 2 | \( 1 + (0.296 + 0.142i)T + (1.24 + 1.56i)T^{2} \) |
| 5 | \( 1 + (0.788 + 3.45i)T + (-4.50 + 2.16i)T^{2} \) |
| 7 | \( 1 + 0.343iT - 7T^{2} \) |
| 11 | \( 1 + (-0.522 - 0.416i)T + (2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (1.40 + 6.15i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (-3.33 - 0.761i)T + (15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 + (3.73 - 2.97i)T + (4.22 - 18.5i)T^{2} \) |
| 23 | \( 1 + (2.07 + 1.65i)T + (5.11 + 22.4i)T^{2} \) |
| 29 | \( 1 + (-4.94 - 2.38i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (-1.59 - 0.766i)T + (19.3 + 24.2i)T^{2} \) |
| 37 | \( 1 + 3.95iT - 37T^{2} \) |
| 41 | \( 1 + (0.752 - 1.56i)T + (-25.5 - 32.0i)T^{2} \) |
| 47 | \( 1 + (-10.4 + 8.32i)T + (10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (6.28 + 1.43i)T + (47.7 + 22.9i)T^{2} \) |
| 59 | \( 1 + (9.98 + 2.27i)T + (53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (0.980 + 2.03i)T + (-38.0 + 47.6i)T^{2} \) |
| 67 | \( 1 + (-5.57 - 6.98i)T + (-14.9 + 65.3i)T^{2} \) |
| 71 | \( 1 + (-4.01 - 5.03i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-8.00 + 1.82i)T + (65.7 - 31.6i)T^{2} \) |
| 79 | \( 1 + 5.14T + 79T^{2} \) |
| 83 | \( 1 + (3.96 + 8.23i)T + (-51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (-1.33 + 0.641i)T + (55.4 - 69.5i)T^{2} \) |
| 97 | \( 1 + (5.36 - 6.73i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−12.70597157180531278908989262488, −12.20186635633515397622114671300, −10.61627740065190102218971383323, −10.03746314914443608781984916323, −8.922942312672871111144042162027, −8.122067369778279573395436731420, −5.81246613963907847720631205488, −5.08676370935997444769217546707, −4.04543979062518329515219604159, −0.70358128174812319309655981754,
2.68525641299357433699683852695, 4.34524171906470915839031719181, 6.33603376354939770718082616966, 7.11914632984077521873772626785, 7.982923505993625113207227904006, 9.380371375941102146618986012711, 10.75991899145717252002024666501, 11.71786962813250591583498890295, 12.36292763364839136680345246924, 13.84105083855115385247977966404
