L(s) = 1 | − 5·5-s − 12.7·7-s − 34.7·11-s − 75.7·13-s − 84.2·17-s + 83.7·19-s − 172.·23-s + 25·25-s + 88.2·29-s − 205.·31-s + 63.7·35-s − 286·37-s − 321.·41-s + 168.·43-s + 390.·47-s − 180.·49-s + 91.6·53-s + 173.·55-s − 122.·59-s − 878.·61-s + 378.·65-s + 1.03e3·67-s − 605.·71-s − 13.8·73-s + 442.·77-s + 573.·79-s + 627.·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.688·7-s − 0.952·11-s − 1.61·13-s − 1.20·17-s + 1.01·19-s − 1.56·23-s + 0.200·25-s + 0.564·29-s − 1.19·31-s + 0.307·35-s − 1.27·37-s − 1.22·41-s + 0.598·43-s + 1.21·47-s − 0.526·49-s + 0.237·53-s + 0.425·55-s − 0.270·59-s − 1.84·61-s + 0.722·65-s + 1.89·67-s − 1.01·71-s − 0.0221·73-s + 0.655·77-s + 0.816·79-s + 0.830·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.2983860910\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2983860910\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
good | 7 | \( 1 + 12.7T + 343T^{2} \) |
| 11 | \( 1 + 34.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 75.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 84.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 83.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 172.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 88.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 205.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 286T + 5.06e4T^{2} \) |
| 41 | \( 1 + 321.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 168.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 390.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 91.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 122.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 878.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.03e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 605.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 13.8T + 3.89e5T^{2} \) |
| 79 | \( 1 - 573.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 627.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
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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
−8.749016548482755205982913833402, −7.77489384298858247854903252706, −7.29576781877854795537829382895, −6.48166615628107344836457939849, −5.43300764934482087129222436746, −4.77455970135566136532238914649, −3.77126314787628267190291023131, −2.82452064750427295288237937925, −2.00073435894270735420874362159, −0.22924545045305409245544262260,
0.22924545045305409245544262260, 2.00073435894270735420874362159, 2.82452064750427295288237937925, 3.77126314787628267190291023131, 4.77455970135566136532238914649, 5.43300764934482087129222436746, 6.48166615628107344836457939849, 7.29576781877854795537829382895, 7.77489384298858247854903252706, 8.749016548482755205982913833402