L(s) = 1 | + (3.36 + 63.9i)2-s + (837. − 837. i)3-s + (−4.07e3 + 429. i)4-s + (5.67e3 − 5.67e3i)5-s + (5.63e4 + 5.06e4i)6-s − 4.58e3·7-s + (−4.11e4 − 2.58e5i)8-s − 8.70e5i·9-s + (3.81e5 + 3.43e5i)10-s + (−1.20e6 − 1.20e6i)11-s + (−3.05e6 + 3.77e6i)12-s + (−3.66e6 − 3.66e6i)13-s + (−1.54e4 − 2.93e5i)14-s − 9.49e6i·15-s + (1.64e7 − 3.50e6i)16-s + 2.38e7·17-s + ⋯ |
L(s) = 1 | + (0.0525 + 0.998i)2-s + (1.14 − 1.14i)3-s + (−0.994 + 0.104i)4-s + (0.362 − 0.362i)5-s + (1.20 + 1.08i)6-s − 0.0389·7-s + (−0.157 − 0.987i)8-s − 1.63i·9-s + (0.381 + 0.343i)10-s + (−0.681 − 0.681i)11-s + (−1.02 + 1.26i)12-s + (−0.759 − 0.759i)13-s + (−0.00204 − 0.0389i)14-s − 0.833i·15-s + (0.977 − 0.208i)16-s + 0.989·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.567 + 0.823i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{13}{2})\) |
\(\approx\) |
\(2.01222 - 1.05766i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.01222 - 1.05766i\) |
\(L(7)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-3.36 - 63.9i)T \) |
good | 3 | \( 1 + (-837. + 837. i)T - 5.31e5iT^{2} \) |
| 5 | \( 1 + (-5.67e3 + 5.67e3i)T - 2.44e8iT^{2} \) |
| 7 | \( 1 + 4.58e3T + 1.38e10T^{2} \) |
| 11 | \( 1 + (1.20e6 + 1.20e6i)T + 3.13e12iT^{2} \) |
| 13 | \( 1 + (3.66e6 + 3.66e6i)T + 2.32e13iT^{2} \) |
| 17 | \( 1 - 2.38e7T + 5.82e14T^{2} \) |
| 19 | \( 1 + (-4.96e7 + 4.96e7i)T - 2.21e15iT^{2} \) |
| 23 | \( 1 + 1.13e8T + 2.19e16T^{2} \) |
| 29 | \( 1 + (-5.30e8 - 5.30e8i)T + 3.53e17iT^{2} \) |
| 31 | \( 1 + 1.47e9iT - 7.87e17T^{2} \) |
| 37 | \( 1 + (1.24e9 - 1.24e9i)T - 6.58e18iT^{2} \) |
| 41 | \( 1 + 7.97e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 + (-3.00e9 - 3.00e9i)T + 3.99e19iT^{2} \) |
| 47 | \( 1 - 1.72e10iT - 1.16e20T^{2} \) |
| 53 | \( 1 + (6.60e9 - 6.60e9i)T - 4.91e20iT^{2} \) |
| 59 | \( 1 + (-2.56e10 - 2.56e10i)T + 1.77e21iT^{2} \) |
| 61 | \( 1 + (-3.97e10 - 3.97e10i)T + 2.65e21iT^{2} \) |
| 67 | \( 1 + (-5.14e10 + 5.14e10i)T - 8.18e21iT^{2} \) |
| 71 | \( 1 - 7.10e10T + 1.64e22T^{2} \) |
| 73 | \( 1 - 1.51e11iT - 2.29e22T^{2} \) |
| 79 | \( 1 - 2.84e11iT - 5.90e22T^{2} \) |
| 83 | \( 1 + (-1.63e10 + 1.63e10i)T - 1.06e23iT^{2} \) |
| 89 | \( 1 + 4.75e11iT - 2.46e23T^{2} \) |
| 97 | \( 1 + 7.25e11T + 6.93e23T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.77524933673738729650585229771, −14.38082135252480570484752096682, −13.49944520436355131510127488195, −12.57121758402452832477546470654, −9.561657396528787289041901063614, −8.212576437735470407874916880441, −7.31020945154223035963913022894, −5.51026206210662001979535176073, −2.98137769574069149048037341250, −0.826534534763405392952726187815,
2.16456664001942397634306708717, 3.43198288593996876095863491466, 4.90227162946286113665542726545, 8.137309367993108381052112465057, 9.805573645086503544880618433667, 10.12801282139365495169990631184, 12.14723333044284284536971141529, 13.98496719224399197703446980563, 14.55083055599028582389734357043, 16.15362920609745782694321437290