L(s) = 1 | + (0.984 + 0.173i)3-s + (0.984 + 1.70i)7-s + (0.939 + 0.342i)9-s + (−1.26 + 0.223i)13-s + (−0.866 + 0.5i)19-s + (0.673 + 1.85i)21-s + (−0.173 − 0.984i)25-s + (0.866 + 0.5i)27-s + (0.342 + 0.592i)31-s + 0.684i·37-s − 1.28·39-s + (1.20 − 1.43i)43-s + (−1.43 + 2.49i)49-s + (−0.939 + 0.342i)57-s + (−0.826 − 0.984i)61-s + ⋯ |
L(s) = 1 | + (0.984 + 0.173i)3-s + (0.984 + 1.70i)7-s + (0.939 + 0.342i)9-s + (−1.26 + 0.223i)13-s + (−0.866 + 0.5i)19-s + (0.673 + 1.85i)21-s + (−0.173 − 0.984i)25-s + (0.866 + 0.5i)27-s + (0.342 + 0.592i)31-s + 0.684i·37-s − 1.28·39-s + (1.20 − 1.43i)43-s + (−1.43 + 2.49i)49-s + (−0.939 + 0.342i)57-s + (−0.826 − 0.984i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.914993294\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.914993294\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.984 - 0.173i)T \) |
| 19 | \( 1 + (0.866 - 0.5i)T \) |
good | 5 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 7 | \( 1 + (-0.984 - 1.70i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (1.26 - 0.223i)T + (0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 29 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 31 | \( 1 + (-0.342 - 0.592i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - 0.684iT - T^{2} \) |
| 41 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 43 | \( 1 + (-1.20 + 1.43i)T + (-0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 53 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 59 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 61 | \( 1 + (0.826 + 0.984i)T + (-0.173 + 0.984i)T^{2} \) |
| 67 | \( 1 + (-0.118 + 0.326i)T + (-0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \) |
| 79 | \( 1 + (-0.342 + 1.93i)T + (-0.939 - 0.342i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 97 | \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \) |
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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
−8.892592645086974520997205683973, −8.138562250291088900445117088816, −7.73329306422025059261537408964, −6.68250286381601383293117084461, −5.79804865595062996231954666028, −4.87785321976588762672413744841, −4.43985412409983088417792327877, −3.16833587641564898889472428469, −2.28821040178392228131470459947, −1.88959673372259977127883812035,
1.01084029191155015134064862048, 2.06318955544343781195272499287, 2.98525072459951675261008068145, 4.21322098383340321972820486590, 4.33916160233273828239404994765, 5.42936839837765639820634858756, 6.78670269627534203568798331236, 7.24210019078269227239138780449, 7.83497400816673519397555842687, 8.334631063099782068671768399298