L(s) = 1 | − 1.31i·5-s + 4.77·7-s + 3·9-s + 2.27i·11-s + 6.09i·13-s − 4.27·17-s − i·19-s + 3.46·23-s + 3.27·25-s + 6.09i·29-s − 2.62·31-s − 6.27i·35-s + 0.837i·37-s − 10.5·41-s − 10.2i·43-s + ⋯ |
L(s) = 1 | − 0.587i·5-s + 1.80·7-s + 9-s + 0.685i·11-s + 1.68i·13-s − 1.03·17-s − 0.229i·19-s + 0.722·23-s + 0.654·25-s + 1.13i·29-s − 0.471·31-s − 1.06i·35-s + 0.137i·37-s − 1.64·41-s − 1.56i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.144418318\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.144418318\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + iT \) |
good | 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 + 1.31iT - 5T^{2} \) |
| 7 | \( 1 - 4.77T + 7T^{2} \) |
| 11 | \( 1 - 2.27iT - 11T^{2} \) |
| 13 | \( 1 - 6.09iT - 13T^{2} \) |
| 17 | \( 1 + 4.27T + 17T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 - 6.09iT - 29T^{2} \) |
| 31 | \( 1 + 2.62T + 31T^{2} \) |
| 37 | \( 1 - 0.837iT - 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 + 10.2iT - 43T^{2} \) |
| 47 | \( 1 + 4.77T + 47T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 + 8.54iT - 59T^{2} \) |
| 61 | \( 1 + 1.31iT - 61T^{2} \) |
| 67 | \( 1 + 8.54iT - 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 - 4.27T + 73T^{2} \) |
| 79 | \( 1 - 4.30T + 79T^{2} \) |
| 83 | \( 1 - 13.0iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 1.45T + 97T^{2} \) |
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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
−9.583023982441507649893715781733, −8.896892215032173817562112540128, −8.290206393357214929101968119481, −7.10086906027559904031150279865, −6.83458887506654892783131065164, −4.99821949383976215696163448755, −4.82568668545160334659386918770, −3.97869620793611312304494692195, −2.00086097701687889257534285920, −1.47828214726312615245377401213,
1.09145196081869241164768145030, 2.32698484820637173328689697976, 3.52271547231018074195822402822, 4.67418640575727778964870878595, 5.29537433827877107004072744788, 6.43260325826271880264155993815, 7.36978594156801280361322015225, 8.075706565316268716914160454711, 8.626279741849461338071419563924, 9.882837459770036902588518980455