L(s) = 1 | + (−1 − 1.73i)5-s + (2.5 + 0.866i)7-s + (−1.5 + 2.59i)11-s + 6·13-s + (−2 + 3.46i)17-s + (−2 − 3.46i)19-s + (2 + 3.46i)23-s + (0.500 − 0.866i)25-s + 5·29-s + (−3.5 + 6.06i)31-s + (−1.00 − 5.19i)35-s − 2·41-s + 8·43-s + (−1 − 1.73i)47-s + (5.5 + 4.33i)49-s + ⋯ |
L(s) = 1 | + (−0.447 − 0.774i)5-s + (0.944 + 0.327i)7-s + (−0.452 + 0.783i)11-s + 1.66·13-s + (−0.485 + 0.840i)17-s + (−0.458 − 0.794i)19-s + (0.417 + 0.722i)23-s + (0.100 − 0.173i)25-s + 0.928·29-s + (−0.628 + 1.08i)31-s + (−0.169 − 0.878i)35-s − 0.312·41-s + 1.21·43-s + (−0.145 − 0.252i)47-s + (0.785 + 0.618i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.790519020\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.790519020\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
good | 5 | \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5 + 8.66i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3 + 5.19i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 15T + 83T^{2} \) |
| 89 | \( 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 5T + 97T^{2} \) |
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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
−9.177803805478452960543132314568, −8.597061421258669667222882239348, −8.160354777585803268312950093429, −7.15234666870312418920262191525, −6.18958289382464225742915066893, −5.16364083859426110812322377308, −4.54989270091038728997487255339, −3.64573618353519178857988394556, −2.17396195477479501661724827651, −1.10755901281562975072652870162,
0.915592310022598177256712504551, 2.39594649831669469681774526345, 3.49481735598403349573924349875, 4.24908049407290959179895575174, 5.36511006869291827397666501949, 6.26793757073321076773438340087, 7.04484795662703222322487643593, 8.032677717087618361400923397047, 8.400638206995269276785213430887, 9.376694052532446034408918411266