L(s) = 1 | + 0.803·5-s + 2.39·7-s − 59.5·11-s + 77.7·13-s + 2.84·17-s + 118.·19-s − 110.·23-s − 124.·25-s + 125.·29-s − 63.0·31-s + 1.92·35-s − 227.·37-s + 324.·41-s + 272.·43-s + 4.93·47-s − 337.·49-s − 598.·53-s − 47.8·55-s + 670.·59-s + 464.·61-s + 62.4·65-s + 769.·67-s + 611.·71-s + 923.·73-s − 142.·77-s − 39.8·79-s − 443.·83-s + ⋯ |
L(s) = 1 | + 0.0718·5-s + 0.129·7-s − 1.63·11-s + 1.65·13-s + 0.0405·17-s + 1.42·19-s − 1.00·23-s − 0.994·25-s + 0.804·29-s − 0.365·31-s + 0.00928·35-s − 1.01·37-s + 1.23·41-s + 0.967·43-s + 0.0153·47-s − 0.983·49-s − 1.55·53-s − 0.117·55-s + 1.47·59-s + 0.975·61-s + 0.119·65-s + 1.40·67-s + 1.02·71-s + 1.48·73-s − 0.210·77-s − 0.0566·79-s − 0.586·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.025272850\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.025272850\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 0.803T + 125T^{2} \) |
| 7 | \( 1 - 2.39T + 343T^{2} \) |
| 11 | \( 1 + 59.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 77.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 2.84T + 4.91e3T^{2} \) |
| 19 | \( 1 - 118.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 110.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 125.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 63.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 227.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 324.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 272.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 4.93T + 1.03e5T^{2} \) |
| 53 | \( 1 + 598.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 670.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 464.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 769.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 611.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 923.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 39.8T + 4.93e5T^{2} \) |
| 83 | \( 1 + 443.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 78.4T + 7.04e5T^{2} \) |
| 97 | \( 1 + 91.0T + 9.12e5T^{2} \) |
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
−9.379672084444096027763302875749, −8.115143953180607255257059295836, −8.018373446194004118376377157147, −6.81409311296282163661734433121, −5.77535894394819124548484104254, −5.27671727657571858877309190209, −4.02516875346954731614892325670, −3.13038401750687970293511796172, −1.99245366402770298080194394048, −0.71857809037027542868175965555,
0.71857809037027542868175965555, 1.99245366402770298080194394048, 3.13038401750687970293511796172, 4.02516875346954731614892325670, 5.27671727657571858877309190209, 5.77535894394819124548484104254, 6.81409311296282163661734433121, 8.018373446194004118376377157147, 8.115143953180607255257059295836, 9.379672084444096027763302875749