L(s) = 1 | + (1.57 + 0.910i)2-s + (−2.34 − 4.05i)4-s + (7.54 − 13.0i)5-s + (16.2 + 8.80i)7-s − 23.0i·8-s + (23.7 − 13.7i)10-s + (8.56 − 4.94i)11-s + 67.8i·13-s + (17.6 + 28.7i)14-s + (2.27 − 3.93i)16-s + (−35.0 − 60.7i)17-s + (−53.2 − 30.7i)19-s − 70.7·20-s + 18.0·22-s + (113. + 65.7i)23-s + ⋯ |
L(s) = 1 | + (0.557 + 0.321i)2-s + (−0.292 − 0.507i)4-s + (0.674 − 1.16i)5-s + (0.879 + 0.475i)7-s − 1.02i·8-s + (0.752 − 0.434i)10-s + (0.234 − 0.135i)11-s + 1.44i·13-s + (0.337 + 0.548i)14-s + (0.0355 − 0.0615i)16-s + (−0.500 − 0.866i)17-s + (−0.642 − 0.371i)19-s − 0.790·20-s + 0.174·22-s + (1.03 + 0.596i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 + 0.505i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.862 + 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.91099 - 0.519098i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.91099 - 0.519098i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-16.2 - 8.80i)T \) |
good | 2 | \( 1 + (-1.57 - 0.910i)T + (4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (-7.54 + 13.0i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-8.56 + 4.94i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 67.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (35.0 + 60.7i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (53.2 + 30.7i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-113. - 65.7i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 158. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (66.2 - 38.2i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (174. - 301. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 138.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 539.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (111. - 193. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-459. + 265. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (271. + 470. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (116. + 67.0i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (160. + 277. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 416. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-472. + 272. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-161. + 279. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 885.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (812. - 1.40e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 739. iT - 9.12e5T^{2} \) |
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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
−14.20038889526236766722753064887, −13.51301170157823363920586685985, −12.39310968018578765366886394918, −11.10703660485721962028160205982, −9.331737968934660585981334036047, −8.864976028133473370378687533655, −6.75449847002380620229600685614, −5.29086332396067042120374576269, −4.58553394522548473168608315268, −1.51051420637737272417373234747,
2.47647433328230877142271468005, 4.05034657919307656217657668826, 5.70081650457084560363137594285, 7.34244631435980309111218919017, 8.571739661729662157842767801327, 10.43052920511272424286099563125, 11.02866613049640481801075135397, 12.51871774896178816480619674564, 13.47647031936171101833299379999, 14.45754611042020139400721981353