| L(s) = 1 | − 3·5-s − 7-s − 3·9-s − 2·11-s + 2·17-s + 3·19-s − 3·23-s + 4·25-s + 29-s + 9·31-s + 3·35-s − 6·37-s − 6·41-s − 9·43-s + 9·45-s − 47-s + 49-s + 5·53-s + 6·55-s + 4·59-s + 6·61-s + 3·63-s + 4·67-s − 10·71-s − 3·73-s + 2·77-s − 3·79-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s − 9-s − 0.603·11-s + 0.485·17-s + 0.688·19-s − 0.625·23-s + 4/5·25-s + 0.185·29-s + 1.61·31-s + 0.507·35-s − 0.986·37-s − 0.937·41-s − 1.37·43-s + 1.34·45-s − 0.145·47-s + 1/7·49-s + 0.686·53-s + 0.809·55-s + 0.520·59-s + 0.768·61-s + 0.377·63-s + 0.488·67-s − 1.18·71-s − 0.351·73-s + 0.227·77-s − 0.337·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4830689120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4830689120\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 15 T + p T^{2} \) |
| 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
−14.00677049521291, −13.55222495539956, −13.17203254219185, −12.22748031772093, −12.08170763848827, −11.64522421823260, −11.25730884288275, −10.44987824604116, −10.15963413012607, −9.577850816325107, −8.790493683326729, −8.254170726610446, −8.136107162256374, −7.473992303430637, −6.848905072684451, −6.419614299501759, −5.512189255578971, −5.292898243731244, −4.501715013654470, −3.904148969292335, −3.219285944619973, −3.000994221263781, −2.150778764351491, −1.107136778838624, −0.2536986665070357,
0.2536986665070357, 1.107136778838624, 2.150778764351491, 3.000994221263781, 3.219285944619973, 3.904148969292335, 4.501715013654470, 5.292898243731244, 5.512189255578971, 6.419614299501759, 6.848905072684451, 7.473992303430637, 8.136107162256374, 8.254170726610446, 8.790493683326729, 9.577850816325107, 10.15963413012607, 10.44987824604116, 11.25730884288275, 11.64522421823260, 12.08170763848827, 12.22748031772093, 13.17203254219185, 13.55222495539956, 14.00677049521291
