L(s) = 1 | − 2·3-s + 9-s − 4·11-s + 2·13-s + 2·19-s − 4·23-s + 4·27-s + 10·29-s + 4·31-s + 8·33-s + 2·37-s − 4·39-s + 12·41-s − 4·43-s − 4·47-s − 2·53-s − 4·57-s + 10·59-s − 6·61-s + 4·67-s + 8·69-s + 12·71-s − 4·73-s + 4·79-s − 11·81-s − 14·83-s − 20·87-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.458·19-s − 0.834·23-s + 0.769·27-s + 1.85·29-s + 0.718·31-s + 1.39·33-s + 0.328·37-s − 0.640·39-s + 1.87·41-s − 0.609·43-s − 0.583·47-s − 0.274·53-s − 0.529·57-s + 1.30·59-s − 0.768·61-s + 0.488·67-s + 0.963·69-s + 1.42·71-s − 0.468·73-s + 0.450·79-s − 1.22·81-s − 1.53·83-s − 2.14·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.052181959\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.052181959\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.80374054424484, −15.44124623971999, −14.54558865043013, −13.93708249659122, −13.56911593983040, −12.69712508781895, −12.46125481999016, −11.75472473891157, −11.25079427374550, −10.82682713116824, −10.12847529745513, −9.879837234909663, −8.915132055420005, −8.104362665540047, −7.970990142175662, −6.894094209463480, −6.466579224390080, −5.769935936436316, −5.368679744330323, −4.675515163975137, −4.087329980244214, −2.995114853896317, −2.503703936317561, −1.275560326830873, −0.4981163886187724,
0.4981163886187724, 1.275560326830873, 2.503703936317561, 2.995114853896317, 4.087329980244214, 4.675515163975137, 5.368679744330323, 5.769935936436316, 6.466579224390080, 6.894094209463480, 7.970990142175662, 8.104362665540047, 8.915132055420005, 9.879837234909663, 10.12847529745513, 10.82682713116824, 11.25079427374550, 11.75472473891157, 12.46125481999016, 12.69712508781895, 13.56911593983040, 13.93708249659122, 14.54558865043013, 15.44124623971999, 15.80374054424484