| L(s) = 1 | + (0.463 + 1.23i)2-s + (−1.72 − 0.135i)3-s + (0.193 − 0.168i)4-s + (0.620 + 0.854i)5-s + (−0.633 − 2.19i)6-s + (2.94 − 1.94i)7-s + (2.62 + 1.41i)8-s + (2.96 + 0.468i)9-s + (−0.768 + 1.16i)10-s + (−1.71 + 1.79i)11-s + (−0.356 + 0.265i)12-s + (−3.67 + 3.51i)13-s + (3.77 + 2.74i)14-s + (−0.956 − 1.56i)15-s + (−0.459 + 3.38i)16-s + (0.363 + 1.11i)17-s + ⋯ |
| L(s) = 1 | + (0.328 + 0.874i)2-s + (−0.996 − 0.0783i)3-s + (0.0966 − 0.0844i)4-s + (0.277 + 0.382i)5-s + (−0.258 − 0.897i)6-s + (1.11 − 0.735i)7-s + (0.927 + 0.499i)8-s + (0.987 + 0.156i)9-s + (−0.242 + 0.368i)10-s + (−0.516 + 0.540i)11-s + (−0.102 + 0.0766i)12-s + (−1.02 + 0.975i)13-s + (1.00 + 0.732i)14-s + (−0.246 − 0.402i)15-s + (−0.114 + 0.847i)16-s + (0.0882 + 0.271i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 - 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15394 + 0.677535i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15394 + 0.677535i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.72 + 0.135i)T \) |
| 71 | \( 1 + (-2.52 + 8.03i)T \) |
| good | 2 | \( 1 + (-0.463 - 1.23i)T + (-1.50 + 1.31i)T^{2} \) |
| 5 | \( 1 + (-0.620 - 0.854i)T + (-1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (-2.94 + 1.94i)T + (2.75 - 6.43i)T^{2} \) |
| 11 | \( 1 + (1.71 - 1.79i)T + (-0.493 - 10.9i)T^{2} \) |
| 13 | \( 1 + (3.67 - 3.51i)T + (0.583 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.363 - 1.11i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.77 - 0.249i)T + (18.6 + 3.39i)T^{2} \) |
| 23 | \( 1 + (-4.92 + 6.17i)T + (-5.11 - 22.4i)T^{2} \) |
| 29 | \( 1 + (3.40 - 1.45i)T + (20.0 - 20.9i)T^{2} \) |
| 31 | \( 1 + (-4.82 + 0.653i)T + (29.8 - 8.24i)T^{2} \) |
| 37 | \( 1 + (5.30 + 6.65i)T + (-8.23 + 36.0i)T^{2} \) |
| 41 | \( 1 + (9.72 - 4.68i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (3.91 + 1.07i)T + (36.9 + 22.0i)T^{2} \) |
| 47 | \( 1 + (8.79 - 5.25i)T + (22.2 - 41.3i)T^{2} \) |
| 53 | \( 1 + (6.64 + 5.80i)T + (7.11 + 52.5i)T^{2} \) |
| 59 | \( 1 + (-2.07 + 0.376i)T + (55.2 - 20.7i)T^{2} \) |
| 61 | \( 1 + (0.168 + 0.111i)T + (23.9 + 56.0i)T^{2} \) |
| 67 | \( 1 + (3.13 + 3.58i)T + (-8.99 + 66.3i)T^{2} \) |
| 73 | \( 1 + (-8.74 + 3.28i)T + (54.9 - 48.0i)T^{2} \) |
| 79 | \( 1 + (2.56 - 4.76i)T + (-43.5 - 65.9i)T^{2} \) |
| 83 | \( 1 + (0.154 + 0.853i)T + (-77.7 + 29.1i)T^{2} \) |
| 89 | \( 1 + (-0.993 + 1.13i)T + (-11.9 - 88.1i)T^{2} \) |
| 97 | \( 1 + (4.01 - 8.34i)T + (-60.4 - 75.8i)T^{2} \) |
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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
−12.48532965112601360474541889603, −11.40653761165736304151181702804, −10.68602668939359972717143497261, −9.914808664198842401830202685289, −8.014102125399042193123629444741, −7.09493059964352493769571951226, −6.52759300199551320768335706761, −5.02663733322240714360378981986, −4.67208798550340966534866660274, −1.83531052128567215995505089157,
1.51513410311852899131424393411, 3.15017804901786765546367841703, 5.13869418417324754902412511921, 5.20393644616552698414551882248, 7.10538133220031921123842138677, 8.115497793675321576072034583944, 9.647294282913627706110152661464, 10.54963465374892427875401720132, 11.54368969364126175854318313966, 11.84613505410415408379358851482
