L(s) = 1 | + (−1.76 − 3.05i)2-s + (−2.23 + 3.87i)4-s + (1.03 + 1.79i)5-s − 12.4·8-s + (3.66 − 6.35i)10-s + (24.5 − 42.5i)11-s + 44.8·13-s + (39.8 + 69.0i)16-s + (−13.2 + 22.9i)17-s + (38.8 + 67.3i)19-s − 9.28·20-s − 173.·22-s + (27.8 + 48.2i)23-s + (60.3 − 104. i)25-s + (−79.1 − 137. i)26-s + ⋯ |
L(s) = 1 | + (−0.624 − 1.08i)2-s + (−0.279 + 0.483i)4-s + (0.0928 + 0.160i)5-s − 0.551·8-s + (0.115 − 0.200i)10-s + (0.674 − 1.16i)11-s + 0.956·13-s + (0.623 + 1.07i)16-s + (−0.189 + 0.327i)17-s + (0.469 + 0.812i)19-s − 0.103·20-s − 1.68·22-s + (0.252 + 0.437i)23-s + (0.482 − 0.836i)25-s + (−0.597 − 1.03i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.299127501\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.299127501\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (1.76 + 3.05i)T + (-4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (-1.03 - 1.79i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-24.5 + 42.5i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 44.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + (13.2 - 22.9i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-38.8 - 67.3i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-27.8 - 48.2i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 121.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-152. + 264. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (38.5 + 66.8i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 248.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 147.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (134. + 233. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (70.5 - 122. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-212. + 367. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (293. + 509. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-89.8 + 155. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 674.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-118. + 205. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (247. + 429. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 24.4T + 5.71e5T^{2} \) |
| 89 | \( 1 + (536. + 928. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.66e3T + 9.12e5T^{2} \) |
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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
−10.44201028403797084971860718531, −9.553155901444702602348769490303, −8.762716224801645747242005340634, −7.983171797757168981164726487399, −6.39023686236648767099978895514, −5.78774246945371347558907728699, −3.94573188379906561813527270681, −3.10121476898386212165224015051, −1.71605578157271568870840825735, −0.60347384408802594537617989148,
1.21410128171113276454091241884, 3.01520101441856137733242150906, 4.51215003737908399113779137706, 5.62336837303069346068379515974, 6.77944201582163996163503680662, 7.17356090786928829198202095625, 8.383507975718510543380885579602, 9.078155758078131526028474425360, 9.759659196481577625635362548784, 10.99748126563437963946081412969