L(s) = 1 | − 3-s − 5-s − 3·7-s + 9-s + 6·11-s − 2·13-s + 15-s + 3·17-s + 6·19-s + 3·21-s + 23-s + 25-s − 27-s − 9·29-s + 3·31-s − 6·33-s + 3·35-s + 3·37-s + 2·39-s − 3·41-s − 45-s − 4·47-s + 2·49-s − 3·51-s − 9·53-s − 6·55-s − 6·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s + 1.80·11-s − 0.554·13-s + 0.258·15-s + 0.727·17-s + 1.37·19-s + 0.654·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.67·29-s + 0.538·31-s − 1.04·33-s + 0.507·35-s + 0.493·37-s + 0.320·39-s − 0.468·41-s − 0.149·45-s − 0.583·47-s + 2/7·49-s − 0.420·51-s − 1.23·53-s − 0.809·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.267219657\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.267219657\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−7.989586710103824428067856864786, −7.29384453127105234509533939558, −6.70902466654960435696911857755, −6.10594516285092160704014512345, −5.34477195253805987643204761809, −4.45076781536427426432080930506, −3.58206158092981118469442472445, −3.15396175658548843645329551116, −1.64627640654696770580664686070, −0.64378858879566636241022575623,
0.64378858879566636241022575623, 1.64627640654696770580664686070, 3.15396175658548843645329551116, 3.58206158092981118469442472445, 4.45076781536427426432080930506, 5.34477195253805987643204761809, 6.10594516285092160704014512345, 6.70902466654960435696911857755, 7.29384453127105234509533939558, 7.989586710103824428067856864786