L(s) = 1 | − 4·5-s + 2·7-s − 5·11-s − 2·13-s + 3·17-s − 19-s − 6·23-s + 11·25-s + 2·29-s + 4·31-s − 8·35-s − 8·37-s − 41-s + 7·43-s + 2·47-s − 3·49-s + 4·53-s + 20·55-s + 5·59-s + 8·65-s + 13·67-s + 8·71-s + 3·73-s − 10·77-s − 8·79-s + 12·83-s − 12·85-s + ⋯ |
L(s) = 1 | − 1.78·5-s + 0.755·7-s − 1.50·11-s − 0.554·13-s + 0.727·17-s − 0.229·19-s − 1.25·23-s + 11/5·25-s + 0.371·29-s + 0.718·31-s − 1.35·35-s − 1.31·37-s − 0.156·41-s + 1.06·43-s + 0.291·47-s − 3/7·49-s + 0.549·53-s + 2.69·55-s + 0.650·59-s + 0.992·65-s + 1.58·67-s + 0.949·71-s + 0.351·73-s − 1.13·77-s − 0.900·79-s + 1.31·83-s − 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8941089179\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8941089179\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−8.358679494795216848701445361575, −8.174575442790596100404529016806, −7.57717790308672452183151903378, −6.87542415992061194770973734581, −5.54749896516649903180227481319, −4.86689590641576688437819758395, −4.13408468526752914925093489745, −3.25609936236852374485306704038, −2.23973121176554611588149032587, −0.57381178061417432178908058149,
0.57381178061417432178908058149, 2.23973121176554611588149032587, 3.25609936236852374485306704038, 4.13408468526752914925093489745, 4.86689590641576688437819758395, 5.54749896516649903180227481319, 6.87542415992061194770973734581, 7.57717790308672452183151903378, 8.174575442790596100404529016806, 8.358679494795216848701445361575