L(s) = 1 | + (1.5 − 2.59i)3-s + (−0.119 − 0.207i)5-s + (−12.9 + 13.2i)7-s + (−4.5 − 7.79i)9-s + (4.04 − 7.01i)11-s + 43.7·13-s − 0.717·15-s + (30.4 − 52.6i)17-s + (−49.4 − 85.6i)19-s + (15.1 + 53.4i)21-s + (−106. − 185. i)23-s + (62.4 − 108. i)25-s − 27·27-s + 110.·29-s + (−40.0 + 69.2i)31-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (−0.0107 − 0.0185i)5-s + (−0.696 + 0.717i)7-s + (−0.166 − 0.288i)9-s + (0.110 − 0.192i)11-s + 0.933·13-s − 0.0123·15-s + (0.433 − 0.751i)17-s + (−0.597 − 1.03i)19-s + (0.157 + 0.555i)21-s + (−0.969 − 1.67i)23-s + (0.499 − 0.865i)25-s − 0.192·27-s + 0.706·29-s + (−0.231 + 0.401i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.439037350\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.439037350\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 + 2.59i)T \) |
| 7 | \( 1 + (12.9 - 13.2i)T \) |
good | 5 | \( 1 + (0.119 + 0.207i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-4.04 + 7.01i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 43.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-30.4 + 52.6i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (49.4 + 85.6i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (106. + 185. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 110.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (40.0 - 69.2i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-1.44 - 2.49i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 242.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 367.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-44.6 - 77.4i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (5.70 - 9.88i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-200. + 346. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (240. + 416. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (120. - 208. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 978.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-311. + 538. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (272. + 472. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 845.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-101. - 176. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.19e3T + 9.12e5T^{2} \) |
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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
−10.87105427113544363562365052110, −9.845560248902488205665694671271, −8.753277359939360500822772599024, −8.277620290197451780438321102046, −6.72978179688841846292069279289, −6.26667424230859767125826747601, −4.84619434292691606971374937160, −3.34241501858837087071526852637, −2.28929889568573236395002451326, −0.50125577844024917112917132801,
1.52808843931193865042588283387, 3.42937756663863946666132343063, 3.98351915981060369519610405419, 5.53766098426031152498719381985, 6.52979278080337394503670768555, 7.71753844160104957964935398910, 8.605811215602421645435358662488, 9.753892718630573581312542085952, 10.29041411790489700947256204474, 11.24112677739310080515664896125