| L(s) = 1 | + (0.841 + 0.540i)2-s + (0.142 + 0.989i)3-s + (0.415 + 0.909i)4-s + (0.959 + 0.281i)5-s + (−0.415 + 0.909i)6-s + (−1.67 + 1.93i)7-s + (−0.142 + 0.989i)8-s + (−0.959 + 0.281i)9-s + (0.654 + 0.755i)10-s + (−2.76 + 1.77i)11-s + (−0.841 + 0.540i)12-s + (−0.430 − 0.497i)13-s + (−2.46 + 0.722i)14-s + (−0.142 + 0.989i)15-s + (−0.654 + 0.755i)16-s + (0.0578 − 0.126i)17-s + ⋯ |
| L(s) = 1 | + (0.594 + 0.382i)2-s + (0.0821 + 0.571i)3-s + (0.207 + 0.454i)4-s + (0.429 + 0.125i)5-s + (−0.169 + 0.371i)6-s + (−0.634 + 0.732i)7-s + (−0.0503 + 0.349i)8-s + (−0.319 + 0.0939i)9-s + (0.207 + 0.238i)10-s + (−0.834 + 0.535i)11-s + (−0.242 + 0.156i)12-s + (−0.119 − 0.137i)13-s + (−0.657 + 0.193i)14-s + (−0.0367 + 0.255i)15-s + (−0.163 + 0.188i)16-s + (0.0140 − 0.0307i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.696670 + 1.71896i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.696670 + 1.71896i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.841 - 0.540i)T \) |
| 3 | \( 1 + (-0.142 - 0.989i)T \) |
| 5 | \( 1 + (-0.959 - 0.281i)T \) |
| 23 | \( 1 + (-0.901 - 4.71i)T \) |
| good | 7 | \( 1 + (1.67 - 1.93i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (2.76 - 1.77i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (0.430 + 0.497i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.0578 + 0.126i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.15 - 2.53i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-1.89 + 4.15i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.109 + 0.764i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-3.95 + 1.16i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (1.26 + 0.370i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.230 - 1.60i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + (1.71 - 1.98i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (2.61 + 3.02i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (0.0859 - 0.597i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-9.14 - 5.87i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (9.47 + 6.08i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (0.989 + 2.16i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-7.89 - 9.11i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (1.26 - 0.372i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (2.22 + 15.4i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-14.2 - 4.19i)T + (81.6 + 52.4i)T^{2} \) |
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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
−10.70352303612831325054057444152, −9.842173425998477741485604650180, −9.229455916940026872982027280038, −8.105846058298374305395879516636, −7.24110565154756468863718982435, −6.01946886031797504796750091877, −5.51179136236131339455518737520, −4.46521506653494528477619141788, −3.24590913196821761647045332235, −2.33684313056035361359163601754,
0.78366116224649735860869976714, 2.39885199969393110607574530579, 3.31852590358669367356131405498, 4.61121522336092013380950195845, 5.62188535492314415898383217522, 6.55749048783568338198884124642, 7.28514586525548406307066788605, 8.425859259265791113124789410016, 9.397572743385600195174111509783, 10.39224113371535395867908693914
