| L(s) = 1 | + (−0.997 + 0.0713i)2-s + (−0.212 + 0.977i)3-s + (0.989 − 0.142i)4-s + (1.62 + 1.53i)5-s + (0.142 − 0.989i)6-s + (1.83 − 3.35i)7-s + (−0.977 + 0.212i)8-s + (−0.909 − 0.415i)9-s + (−1.73 − 1.41i)10-s + (1.05 − 0.913i)11-s + (−0.0713 + 0.997i)12-s + (4.41 − 2.41i)13-s + (−1.58 + 3.47i)14-s + (−1.84 + 1.26i)15-s + (0.959 − 0.281i)16-s + (−4.62 − 6.18i)17-s + ⋯ |
| L(s) = 1 | + (−0.705 + 0.0504i)2-s + (−0.122 + 0.564i)3-s + (0.494 − 0.0711i)4-s + (0.726 + 0.686i)5-s + (0.0580 − 0.404i)6-s + (0.692 − 1.26i)7-s + (−0.345 + 0.0751i)8-s + (−0.303 − 0.138i)9-s + (−0.547 − 0.447i)10-s + (0.317 − 0.275i)11-s + (−0.0205 + 0.287i)12-s + (1.22 − 0.668i)13-s + (−0.424 + 0.929i)14-s + (−0.476 + 0.325i)15-s + (0.239 − 0.0704i)16-s + (−1.12 − 1.49i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.25452 - 0.192946i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25452 - 0.192946i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.62 - 1.53i)T \) |
| 23 | \( 1 + (-4.78 - 0.367i)T \) |
| good | 7 | \( 1 + (-1.83 + 3.35i)T + (-3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-1.05 + 0.913i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-4.41 + 2.41i)T + (7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (4.62 + 6.18i)T + (-4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (0.753 + 5.23i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (5.88 + 0.846i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (8.20 + 5.27i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-0.492 - 1.31i)T + (-27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (-2.71 - 5.94i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-4.16 - 0.905i)T + (39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-7.10 - 7.10i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.49 - 1.36i)T + (28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (1.11 - 3.78i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (1.03 - 1.60i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (0.824 + 11.5i)T + (-66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (2.57 - 2.96i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-6.27 - 4.69i)T + (20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-7.27 - 2.13i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-8.24 + 3.07i)T + (62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (0.749 - 0.481i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (4.59 + 1.71i)T + (73.3 + 63.5i)T^{2} \) |
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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
−10.73648781192558874595441375726, −9.335221370111490564636245338404, −9.141008355893616543484380208810, −7.71267097705688706854042942355, −7.04007005823928014172194997884, −6.11647872077504914537643773978, −4.99988272366741719766450579544, −3.80471651239714524641493007416, −2.57948994342403338050568595192, −0.927410715352264593738972344606,
1.64384597934323610388390166566, 1.99445749027928597663110543422, 3.95269273050408049347657386243, 5.48198050939191037787985831032, 6.00081733558731831390665834035, 6.98681867093387112566589535411, 8.309857228216964431910947741010, 8.797566740888039853465663123279, 9.231635442942625862649445499077, 10.67201103400112314076507322915
