| L(s) = 1 | + (−0.142 + 0.989i)2-s + (−0.415 − 0.909i)3-s + (−0.959 − 0.281i)4-s + (−0.654 − 0.755i)5-s + (0.959 − 0.281i)6-s + (1.99 + 1.28i)7-s + (0.415 − 0.909i)8-s + (−0.654 + 0.755i)9-s + (0.841 − 0.540i)10-s + (−0.0482 − 0.335i)11-s + (0.142 + 0.989i)12-s + (−3.00 + 1.93i)13-s + (−1.55 + 1.79i)14-s + (−0.415 + 0.909i)15-s + (0.841 + 0.540i)16-s + (5.91 − 1.73i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (−0.239 − 0.525i)3-s + (−0.479 − 0.140i)4-s + (−0.292 − 0.337i)5-s + (0.391 − 0.115i)6-s + (0.755 + 0.485i)7-s + (0.146 − 0.321i)8-s + (−0.218 + 0.251i)9-s + (0.266 − 0.170i)10-s + (−0.0145 − 0.101i)11-s + (0.0410 + 0.285i)12-s + (−0.834 + 0.536i)13-s + (−0.416 + 0.480i)14-s + (−0.107 + 0.234i)15-s + (0.210 + 0.135i)16-s + (1.43 − 0.421i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.28180 + 0.147911i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28180 + 0.147911i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 3 | \( 1 + (0.415 + 0.909i)T \) |
| 5 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (0.240 + 4.78i)T \) |
| good | 7 | \( 1 + (-1.99 - 1.28i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (0.0482 + 0.335i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (3.00 - 1.93i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-5.91 + 1.73i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-3.66 - 1.07i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-8.33 + 2.44i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-2.08 + 4.56i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (0.647 - 0.747i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-0.803 - 0.927i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-2.71 - 5.95i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 9.10T + 47T^{2} \) |
| 53 | \( 1 + (-5.84 - 3.75i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-1.65 + 1.06i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-0.831 + 1.82i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (1.28 - 8.91i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.857 + 5.96i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (4.05 + 1.19i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (6.04 - 3.88i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (3.14 - 3.62i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (5.32 + 11.6i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-2.50 - 2.88i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.35395725062826852476199925178, −9.509304884638368647539646330858, −8.483571856601203955507174790645, −7.85178992374864712490131142189, −7.13810130988736454577485198136, −6.01081947478367875024950775460, −5.19560076775113686257579142355, −4.36637478406418372855642329295, −2.64835687334668487110938428351, −1.00310266274271398998255380383,
1.10931849327668575218135234670, 2.84238517762667612600727183322, 3.77928825643567378643198332498, 4.86428305249084833254315951995, 5.59256328825512192417475486663, 7.19057468231365945989974664550, 7.84688005147564670383407356872, 8.842203877453834822534720221570, 10.03690300269783119354278526313, 10.29503590782753152033727208552
