L(s) = 1 | + (1.07 − 0.921i)2-s − 3-s + (0.303 − 1.97i)4-s − i·5-s + (−1.07 + 0.921i)6-s + (−1.82 + 1.91i)7-s + (−1.49 − 2.40i)8-s + 9-s + (−0.921 − 1.07i)10-s − 6.24i·11-s + (−0.303 + 1.97i)12-s − 2.40i·13-s + (−0.195 + 3.73i)14-s + i·15-s + (−3.81 − 1.19i)16-s − 1.30i·17-s + ⋯ |
L(s) = 1 | + (0.758 − 0.651i)2-s − 0.577·3-s + (0.151 − 0.988i)4-s − 0.447i·5-s + (−0.438 + 0.376i)6-s + (−0.690 + 0.723i)7-s + (−0.528 − 0.848i)8-s + 0.333·9-s + (−0.291 − 0.339i)10-s − 1.88i·11-s + (−0.0875 + 0.570i)12-s − 0.666i·13-s + (−0.0522 + 0.998i)14-s + 0.258i·15-s + (−0.954 − 0.299i)16-s − 0.317i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.791 + 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.791 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.436363 - 1.28012i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.436363 - 1.28012i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.07 + 0.921i)T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (1.82 - 1.91i)T \) |
good | 11 | \( 1 + 6.24iT - 11T^{2} \) |
| 13 | \( 1 + 2.40iT - 13T^{2} \) |
| 17 | \( 1 + 1.30iT - 17T^{2} \) |
| 19 | \( 1 + 3.94T + 19T^{2} \) |
| 23 | \( 1 - 3.55iT - 23T^{2} \) |
| 29 | \( 1 - 1.44T + 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 + 5.06T + 37T^{2} \) |
| 41 | \( 1 - 1.26iT - 41T^{2} \) |
| 43 | \( 1 - 2.19iT - 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 - 0.415T + 59T^{2} \) |
| 61 | \( 1 + 12.6iT - 61T^{2} \) |
| 67 | \( 1 - 3.10iT - 67T^{2} \) |
| 71 | \( 1 + 10.4iT - 71T^{2} \) |
| 73 | \( 1 + 1.64iT - 73T^{2} \) |
| 79 | \( 1 - 9.18iT - 79T^{2} \) |
| 83 | \( 1 - 7.39T + 83T^{2} \) |
| 89 | \( 1 - 11.4iT - 89T^{2} \) |
| 97 | \( 1 - 12.4iT - 97T^{2} \) |
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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
−10.99354931020380424200495714661, −10.22977465668812966658230433753, −9.184691327175789624024940689273, −8.291278838789381163360574777571, −6.54246164999725178924137314122, −5.86046309949693312771149514465, −5.13778107847059483416697712625, −3.74329754449554148677176565167, −2.68646771423661718318265400103, −0.72560773242576735872262237443,
2.39829461921330773233093892289, 4.06860985016149564162610863210, 4.59801654155966654415282620550, 6.05656420805541778607505040600, 6.89161372454685217135653301260, 7.27233783324670151456862498132, 8.673470470502829704388546327543, 9.999345598862762552524287027156, 10.58472031563445936959941542337, 11.92761870230492976417053142243