| L(s) = 1 | + (−0.847 − 1.13i)2-s + (−0.562 + 1.91i)4-s − 2.55i·5-s + 3.08i·7-s + (2.64 − 0.990i)8-s + (−2.88 + 2.16i)10-s − 5.76·11-s + 1.95·13-s + (3.49 − 2.61i)14-s + (−3.36 − 2.15i)16-s − 1.39·17-s + (3.64 + 2.39i)19-s + (4.89 + 1.43i)20-s + (4.88 + 6.52i)22-s − 1.59i·23-s + ⋯ |
| L(s) = 1 | + (−0.599 − 0.800i)2-s + (−0.281 + 0.959i)4-s − 1.14i·5-s + 1.16i·7-s + (0.936 − 0.350i)8-s + (−0.913 + 0.684i)10-s − 1.73·11-s + 0.541·13-s + (0.934 − 0.700i)14-s + (−0.841 − 0.539i)16-s − 0.339·17-s + (0.835 + 0.550i)19-s + (1.09 + 0.321i)20-s + (1.04 + 1.39i)22-s − 0.331i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9764932993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9764932993\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.847 + 1.13i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-3.64 - 2.39i)T \) |
| good | 5 | \( 1 + 2.55iT - 5T^{2} \) |
| 7 | \( 1 - 3.08iT - 7T^{2} \) |
| 11 | \( 1 + 5.76T + 11T^{2} \) |
| 13 | \( 1 - 1.95T + 13T^{2} \) |
| 17 | \( 1 + 1.39T + 17T^{2} \) |
| 23 | \( 1 + 1.59iT - 23T^{2} \) |
| 29 | \( 1 - 7.86T + 29T^{2} \) |
| 31 | \( 1 + 4.85T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 - 6.86iT - 41T^{2} \) |
| 43 | \( 1 + 8.37T + 43T^{2} \) |
| 47 | \( 1 - 8.93iT - 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 - 12.8iT - 59T^{2} \) |
| 61 | \( 1 - 6.56iT - 61T^{2} \) |
| 67 | \( 1 + 16.2iT - 67T^{2} \) |
| 71 | \( 1 + 4.47T + 71T^{2} \) |
| 73 | \( 1 - 3.48T + 73T^{2} \) |
| 79 | \( 1 - 8.99T + 79T^{2} \) |
| 83 | \( 1 - 8.73T + 83T^{2} \) |
| 89 | \( 1 + 9.05iT - 89T^{2} \) |
| 97 | \( 1 - 14.6iT - 97T^{2} \) |
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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
−9.481904701418581974616242808381, −8.790219986127895271631503489015, −8.217247905620567640433092531922, −7.62078948007269844532624415303, −6.10950099703221769146208193199, −5.17476525018742033732323361541, −4.52479184496592774757137137673, −3.09284296591765155640923536482, −2.30606770043114376760375542742, −0.987086401795242225798101677221,
0.62691540735607244703883542740, 2.36281481860669288362328027048, 3.52405920233241862773710562176, 4.76206421046010888566283911733, 5.60714396781906547206670173950, 6.67982321413601305048950806551, 7.18404519124070290954095174181, 7.82019934514951631022452007540, 8.600853284885047774669115422938, 9.816585172295265272516942967493
