L(s) = 1 | + (−2.60 − 4.50i)2-s + (−9.52 + 16.4i)4-s + (2.5 − 4.33i)5-s + (−12.2 − 21.1i)7-s + 57.4·8-s − 26.0·10-s + (14.4 + 25.1i)11-s + (32.6 − 56.6i)13-s + (−63.4 + 109. i)14-s + (−73.2 − 126. i)16-s + 68.1·17-s + 104.·19-s + (47.6 + 82.4i)20-s + (75.3 − 130. i)22-s + (77.4 − 134. i)23-s + ⋯ |
L(s) = 1 | + (−0.919 − 1.59i)2-s + (−1.19 + 2.06i)4-s + (0.223 − 0.387i)5-s + (−0.658 − 1.14i)7-s + 2.53·8-s − 0.822·10-s + (0.397 + 0.688i)11-s + (0.697 − 1.20i)13-s + (−1.21 + 2.09i)14-s + (−1.14 − 1.98i)16-s + 0.972·17-s + 1.26·19-s + (0.532 + 0.922i)20-s + (0.730 − 1.26i)22-s + (0.701 − 1.21i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.002089122\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.002089122\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.5 + 4.33i)T \) |
good | 2 | \( 1 + (2.60 + 4.50i)T + (-4 + 6.92i)T^{2} \) |
| 7 | \( 1 + (12.2 + 21.1i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-14.4 - 25.1i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-32.6 + 56.6i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 68.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 104.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-77.4 + 134. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-102. - 178. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-9.12 + 15.8i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 337.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-97.9 + 169. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (167. + 290. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-2.50 - 4.33i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 319.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (215. - 372. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (297. + 514. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (97.9 - 169. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 425.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 929.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (12.2 + 21.1i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-272. - 472. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 84.1T + 7.04e5T^{2} \) |
| 97 | \( 1 + (413. + 716. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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
−10.37453902342126574490060844370, −9.711296167230333953018142768467, −8.831455286732166945351144675053, −7.87663347213059101267569251599, −6.89846408083559547157506040530, −5.10519808507630973869623026683, −3.74721732593318381586556115251, −3.05709744972290142118562456616, −1.35922898951436535848849845228, −0.57117780841069987616588325583,
1.29243463880616166966207908099, 3.30334322728650956723045256184, 5.15891709993940923021704398767, 6.02493158310532974860156963541, 6.55248794447728205832758598794, 7.61644660386908196227482523217, 8.584656048308783637967954689644, 9.402347463907572586629620835572, 9.750778100917915478845383864481, 11.17433988923134811903825828855