L(s) = 1 | + (−1.26 − 0.639i)2-s + (1.57 − 0.730i)3-s + (1.18 + 1.61i)4-s + (−2.44 − 0.0838i)6-s + 1.25i·7-s + (−0.458 − 2.79i)8-s + (1.93 − 2.29i)9-s + 3.02i·11-s + (3.03 + 1.67i)12-s − 5.65i·13-s + (0.803 − 1.58i)14-s + (−1.20 + 3.81i)16-s − 2.45i·17-s + (−3.90 + 1.65i)18-s + 1.77·19-s + ⋯ |
L(s) = 1 | + (−0.891 − 0.452i)2-s + (0.906 − 0.421i)3-s + (0.590 + 0.806i)4-s + (−0.999 − 0.0342i)6-s + 0.474i·7-s + (−0.162 − 0.986i)8-s + (0.644 − 0.764i)9-s + 0.911i·11-s + (0.875 + 0.482i)12-s − 1.56i·13-s + (0.214 − 0.423i)14-s + (−0.301 + 0.953i)16-s − 0.595i·17-s + (−0.920 + 0.390i)18-s + 0.407·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.22940 - 0.650198i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.22940 - 0.650198i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.26 + 0.639i)T \) |
| 3 | \( 1 + (-1.57 + 0.730i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 1.25iT - 7T^{2} \) |
| 11 | \( 1 - 3.02iT - 11T^{2} \) |
| 13 | \( 1 + 5.65iT - 13T^{2} \) |
| 17 | \( 1 + 2.45iT - 17T^{2} \) |
| 19 | \( 1 - 1.77T + 19T^{2} \) |
| 23 | \( 1 - 8.84T + 23T^{2} \) |
| 29 | \( 1 - 3.79T + 29T^{2} \) |
| 31 | \( 1 + 5.19iT - 31T^{2} \) |
| 37 | \( 1 - 6.45iT - 37T^{2} \) |
| 41 | \( 1 - 7.57iT - 41T^{2} \) |
| 43 | \( 1 - 4.37T + 43T^{2} \) |
| 47 | \( 1 + 1.83T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 - 4.91iT - 59T^{2} \) |
| 61 | \( 1 + 8.16iT - 61T^{2} \) |
| 67 | \( 1 + 8.50T + 67T^{2} \) |
| 71 | \( 1 - 7.00T + 71T^{2} \) |
| 73 | \( 1 + 4.59T + 73T^{2} \) |
| 79 | \( 1 + 7.36iT - 79T^{2} \) |
| 83 | \( 1 + 15.7iT - 83T^{2} \) |
| 89 | \( 1 - 3.65iT - 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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
−10.26033547286019056299277244302, −9.565467481603088898762192134630, −8.865161207633759064579637682693, −7.918470548954518865218503980638, −7.40051820661082975543902653000, −6.36594288814599235565937853941, −4.79563033949099451749809273182, −3.20095950132259066413469768720, −2.60270675642621597050104604013, −1.12266409584375841745951544379,
1.42509976847702703621453727802, 2.86297690941946191305457273602, 4.15674258839741793705206751431, 5.34545552269549045395924448768, 6.69857139569296209386594116166, 7.31621396706233110910727281869, 8.418433987249476131660959547894, 8.977674666351312751938949824953, 9.628452369028765343096274093129, 10.76388803884973381829889797169