| L(s) = 1 | + (−0.0713 + 0.997i)2-s + (0.977 − 0.212i)3-s + (−0.989 − 0.142i)4-s + (2.11 + 0.723i)5-s + (0.142 + 0.989i)6-s + (1.77 − 0.968i)7-s + (0.212 − 0.977i)8-s + (0.909 − 0.415i)9-s + (−0.872 + 2.05i)10-s + (0.321 + 0.278i)11-s + (−0.997 + 0.0713i)12-s + (2.55 − 4.67i)13-s + (0.839 + 1.83i)14-s + (2.22 + 0.257i)15-s + (0.959 + 0.281i)16-s + (−0.995 − 0.744i)17-s + ⋯ |
| L(s) = 1 | + (−0.0504 + 0.705i)2-s + (0.564 − 0.122i)3-s + (−0.494 − 0.0711i)4-s + (0.946 + 0.323i)5-s + (0.0580 + 0.404i)6-s + (0.670 − 0.365i)7-s + (0.0751 − 0.345i)8-s + (0.303 − 0.138i)9-s + (−0.275 + 0.651i)10-s + (0.0969 + 0.0840i)11-s + (−0.287 + 0.0205i)12-s + (0.707 − 1.29i)13-s + (0.224 + 0.491i)14-s + (0.573 + 0.0664i)15-s + (0.239 + 0.0704i)16-s + (−0.241 − 0.180i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.05890 + 0.614502i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.05890 + 0.614502i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0713 - 0.997i)T \) |
| 3 | \( 1 + (-0.977 + 0.212i)T \) |
| 5 | \( 1 + (-2.11 - 0.723i)T \) |
| 23 | \( 1 + (4.53 - 1.55i)T \) |
| good | 7 | \( 1 + (-1.77 + 0.968i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-0.321 - 0.278i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.55 + 4.67i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (0.995 + 0.744i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-0.0697 + 0.485i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.244 + 0.0351i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (1.06 - 0.682i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (1.57 + 0.587i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (0.149 - 0.326i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-0.634 - 2.91i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (0.514 + 0.514i)T + 47iT^{2} \) |
| 53 | \( 1 + (-6.27 - 11.4i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (-2.87 - 9.79i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (7.18 + 11.1i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-6.42 - 0.459i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-6.34 - 7.32i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (3.98 + 5.32i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (12.0 - 3.54i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (3.08 - 8.26i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (12.7 + 8.17i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-1.30 - 3.50i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−10.37287929428696671054701515272, −9.603907375854271146149210641864, −8.672167539837199154103189752782, −7.928799108430975086831760408506, −7.12616891348510842842829531110, −6.09606442321137937294462470006, −5.33950630669292709323213729087, −4.11300948747896514721632510739, −2.84320263787007680313059497898, −1.39154313937231367002767722954,
1.59669155758798327596296999241, 2.29108690287366182767708013161, 3.77925614145792985803905293218, 4.70364023500835811263043317765, 5.76342116755712762464937485437, 6.81846875115386256386311695800, 8.283967389705524317814684186778, 8.743817087795260180954014158290, 9.537113516838017999000345198888, 10.27757786407330281790856670341
