| L(s) = 1 | + (0.955 + 0.294i)2-s + (0.365 − 0.930i)3-s + (0.826 + 0.563i)4-s + (1.22 − 0.184i)5-s + (0.623 − 0.781i)6-s + (0.792 + 2.52i)7-s + (0.623 + 0.781i)8-s + (−0.733 − 0.680i)9-s + (1.22 + 0.184i)10-s + (−0.146 + 0.136i)11-s + (0.826 − 0.563i)12-s + (−0.348 − 1.52i)13-s + (0.0129 + 2.64i)14-s + (0.275 − 1.20i)15-s + (0.365 + 0.930i)16-s + (−0.278 − 3.71i)17-s + ⋯ |
| L(s) = 1 | + (0.675 + 0.208i)2-s + (0.210 − 0.537i)3-s + (0.413 + 0.281i)4-s + (0.546 − 0.0824i)5-s + (0.254 − 0.319i)6-s + (0.299 + 0.954i)7-s + (0.220 + 0.276i)8-s + (−0.244 − 0.226i)9-s + (0.386 + 0.0582i)10-s + (−0.0442 + 0.0410i)11-s + (0.238 − 0.162i)12-s + (−0.0965 − 0.422i)13-s + (0.00344 + 0.707i)14-s + (0.0710 − 0.311i)15-s + (0.0913 + 0.232i)16-s + (−0.0675 − 0.901i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.14948 + 0.0564371i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.14948 + 0.0564371i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.955 - 0.294i)T \) |
| 3 | \( 1 + (-0.365 + 0.930i)T \) |
| 7 | \( 1 + (-0.792 - 2.52i)T \) |
| good | 5 | \( 1 + (-1.22 + 0.184i)T + (4.77 - 1.47i)T^{2} \) |
| 11 | \( 1 + (0.146 - 0.136i)T + (0.822 - 10.9i)T^{2} \) |
| 13 | \( 1 + (0.348 + 1.52i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (0.278 + 3.71i)T + (-16.8 + 2.53i)T^{2} \) |
| 19 | \( 1 + (-0.160 + 0.278i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0653 + 0.871i)T + (-22.7 - 3.42i)T^{2} \) |
| 29 | \( 1 + (-1.09 - 0.527i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (0.447 + 0.774i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (8.45 - 5.76i)T + (13.5 - 34.4i)T^{2} \) |
| 41 | \( 1 + (3.56 + 4.47i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (3.05 - 3.82i)T + (-9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (-11.3 - 3.51i)T + (38.8 + 26.4i)T^{2} \) |
| 53 | \( 1 + (10.0 + 6.82i)T + (19.3 + 49.3i)T^{2} \) |
| 59 | \( 1 + (6.38 + 0.961i)T + (56.3 + 17.3i)T^{2} \) |
| 61 | \( 1 + (-1.41 + 0.963i)T + (22.2 - 56.7i)T^{2} \) |
| 67 | \( 1 + (-0.872 - 1.51i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-9.28 + 4.47i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-3.27 + 1.01i)T + (60.3 - 41.1i)T^{2} \) |
| 79 | \( 1 + (6.88 - 11.9i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.72 + 7.55i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-5.99 - 5.56i)T + (6.65 + 88.7i)T^{2} \) |
| 97 | \( 1 + 7.14T + 97T^{2} \) |
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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
−12.04948897356111062081861186433, −11.15056692654538480720881166610, −9.820847550338669037718843703341, −8.804187384645220036667619954995, −7.84897309434771811525895870888, −6.74561903549924707649030844928, −5.73052620092018629601062605055, −4.91027739313006581898516060477, −3.13879861834960326997086358480, −1.98459232975592078703444645668,
1.89181267320996496993699551276, 3.52809220064463571121965434591, 4.44506362397856497344029810281, 5.58942924435584428552994276343, 6.71208978617497143414079358771, 7.85583459164919557532735232361, 9.105141719274132862195741227237, 10.21316823630971351921389465338, 10.68451470581181171839444966030, 11.73713080579366785631965024983
