L(s) = 1 | + (−2.83 + 4.35i)3-s + (−2.97 + 5.16i)5-s + (17.8 + 5.06i)7-s + (−10.9 − 24.6i)9-s + (49.9 − 28.8i)11-s − 52.0i·13-s + (−14.0 − 27.5i)15-s + (−17.3 − 30.1i)17-s + (26.7 + 15.4i)19-s + (−72.5 + 63.2i)21-s + (21.8 + 12.6i)23-s + (44.7 + 77.5i)25-s + (138. + 22.0i)27-s + 64.9i·29-s + (80.6 − 46.5i)31-s + ⋯ |
L(s) = 1 | + (−0.544 + 0.838i)3-s + (−0.266 + 0.461i)5-s + (0.961 + 0.273i)7-s + (−0.406 − 0.913i)9-s + (1.36 − 0.789i)11-s − 1.11i·13-s + (−0.241 − 0.474i)15-s + (−0.247 − 0.429i)17-s + (0.322 + 0.186i)19-s + (−0.753 + 0.657i)21-s + (0.198 + 0.114i)23-s + (0.357 + 0.620i)25-s + (0.987 + 0.157i)27-s + 0.415i·29-s + (0.467 − 0.269i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.741764960\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.741764960\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.83 - 4.35i)T \) |
| 7 | \( 1 + (-17.8 - 5.06i)T \) |
good | 5 | \( 1 + (2.97 - 5.16i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-49.9 + 28.8i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 52.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (17.3 + 30.1i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-26.7 - 15.4i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-21.8 - 12.6i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 64.9iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-80.6 + 46.5i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-162. + 280. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 282.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 441.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (291. - 505. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-199. + 115. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (88.3 + 153. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-589. - 340. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (56.2 + 97.4i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 705. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (240. - 138. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-275. + 476. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.12e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-643. + 1.11e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 76.4iT - 9.12e5T^{2} \) |
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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
−11.31963160861017724163006272610, −10.49604665251534179180410940891, −9.351818950574542825419415289209, −8.573158925054341119303547718958, −7.37893159845072105149572963838, −6.13205210603169363710719801498, −5.27835414697960908474856413286, −4.11553026903401072873056553533, −3.05393252445426125298774689213, −0.939764862628316490733659758740,
1.03218570732649646309503637028, 1.99614206341492553897732456848, 4.20208764387856022462972077214, 4.92717404432347197293023054485, 6.40855783336997419673429091751, 7.06854791236644945700911471161, 8.158482286850069482974587320764, 8.966702550551555933262947865764, 10.23349467075260220765622131052, 11.53803008998518714461504026720