| L(s) = 1 | + (0.997 + 0.0713i)2-s + (0.212 + 0.977i)3-s + (0.989 + 0.142i)4-s + (−2.18 − 0.455i)5-s + (0.142 + 0.989i)6-s + (0.172 + 0.316i)7-s + (0.977 + 0.212i)8-s + (−0.909 + 0.415i)9-s + (−2.15 − 0.610i)10-s + (4.35 + 3.77i)11-s + (0.0713 + 0.997i)12-s + (0.701 + 0.383i)13-s + (0.149 + 0.328i)14-s + (−0.0202 − 2.23i)15-s + (0.959 + 0.281i)16-s + (0.276 − 0.369i)17-s + ⋯ |
| L(s) = 1 | + (0.705 + 0.0504i)2-s + (0.122 + 0.564i)3-s + (0.494 + 0.0711i)4-s + (−0.979 − 0.203i)5-s + (0.0580 + 0.404i)6-s + (0.0653 + 0.119i)7-s + (0.345 + 0.0751i)8-s + (−0.303 + 0.138i)9-s + (−0.680 − 0.193i)10-s + (1.31 + 1.13i)11-s + (0.0205 + 0.287i)12-s + (0.194 + 0.106i)13-s + (0.0400 + 0.0877i)14-s + (−0.00522 − 0.577i)15-s + (0.239 + 0.0704i)16-s + (0.0670 − 0.0895i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.64945 + 1.27500i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64945 + 1.27500i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.997 - 0.0713i)T \) |
| 3 | \( 1 + (-0.212 - 0.977i)T \) |
| 5 | \( 1 + (2.18 + 0.455i)T \) |
| 23 | \( 1 + (-2.21 - 4.25i)T \) |
| good | 7 | \( 1 + (-0.172 - 0.316i)T + (-3.78 + 5.88i)T^{2} \) |
| 11 | \( 1 + (-4.35 - 3.77i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.701 - 0.383i)T + (7.02 + 10.9i)T^{2} \) |
| 17 | \( 1 + (-0.276 + 0.369i)T + (-4.78 - 16.3i)T^{2} \) |
| 19 | \( 1 + (0.891 - 6.20i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (9.47 - 1.36i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-1.31 + 0.845i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-3.10 + 8.32i)T + (-27.9 - 24.2i)T^{2} \) |
| 41 | \( 1 + (1.67 - 3.66i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-7.03 + 1.53i)T + (39.1 - 17.8i)T^{2} \) |
| 47 | \( 1 + (1.40 - 1.40i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.55 - 0.849i)T + (28.6 - 44.5i)T^{2} \) |
| 59 | \( 1 + (4.03 + 13.7i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (4.25 + 6.61i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (0.842 - 11.7i)T + (-66.3 - 9.53i)T^{2} \) |
| 71 | \( 1 + (2.46 + 2.84i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-12.9 + 9.68i)T + (20.5 - 70.0i)T^{2} \) |
| 79 | \( 1 + (-3.46 + 1.01i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (0.718 + 0.268i)T + (62.7 + 54.3i)T^{2} \) |
| 89 | \( 1 + (3.65 + 2.35i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (0.902 - 0.336i)T + (73.3 - 63.5i)T^{2} \) |
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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
−10.91947697932143964300733665172, −9.693927401856974943982935431740, −9.054382583823036688665448233798, −7.86683604969447486574942159014, −7.20049239955022943389633410924, −6.05573884163681012049705804077, −4.98298491574745481521925971870, −4.00066828180382995504168536972, −3.57843080555454696577061230271, −1.78918477723248997999996894893,
0.945767247636509829174207764129, 2.74450621140112584503102347781, 3.68927754547094543357548767274, 4.57674452214905606462189950431, 5.95314223310508593670156646837, 6.71799819688249314062476716843, 7.47627467859083192282195369206, 8.496778546621108518431258172488, 9.185863910970880344008137967562, 10.79782947001384546585031847841
