L(s) = 1 | + 13.2·5-s − 5.23·7-s − 11-s + 84.9·13-s − 40.9·17-s + 57.5·19-s − 114.·23-s + 50.1·25-s + 202.·29-s + 274.·31-s − 69.2·35-s + 242.·37-s + 328.·41-s + 281.·43-s − 23.8·47-s − 315.·49-s − 300.·53-s − 13.2·55-s − 753.·59-s + 495.·61-s + 1.12e3·65-s − 409.·67-s − 1.11e3·71-s − 287·73-s + 5.23·77-s − 1.23e3·79-s + 942.·83-s + ⋯ |
L(s) = 1 | + 1.18·5-s − 0.282·7-s − 0.0274·11-s + 1.81·13-s − 0.584·17-s + 0.694·19-s − 1.04·23-s + 0.401·25-s + 1.29·29-s + 1.58·31-s − 0.334·35-s + 1.07·37-s + 1.25·41-s + 0.997·43-s − 0.0740·47-s − 0.920·49-s − 0.777·53-s − 0.0324·55-s − 1.66·59-s + 1.03·61-s + 2.14·65-s − 0.747·67-s − 1.86·71-s − 0.460·73-s + 0.00774·77-s − 1.75·79-s + 1.24·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.234462073\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.234462073\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 - 13.2T + 125T^{2} \) |
| 7 | \( 1 + 5.23T + 343T^{2} \) |
| 11 | \( 1 + T + 1.33e3T^{2} \) |
| 13 | \( 1 - 84.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 40.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 57.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 114.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 202.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 274.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 242.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 328.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 281.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 23.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 300.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 753.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 495.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 409.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.11e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 287T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.23e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 942.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 190.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 306.T + 9.12e5T^{2} \) |
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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
−11.83835515331070642536891061116, −10.78393469598445035558312701337, −9.905074771040248132470443821665, −9.031588197240277705946068079454, −7.982197569423095370460507509881, −6.34620304303927934957474450570, −5.95219174120883752919284721674, −4.35880564921701000208727633144, −2.80275563037754282454914468864, −1.27077781699008545401896784942,
1.27077781699008545401896784942, 2.80275563037754282454914468864, 4.35880564921701000208727633144, 5.95219174120883752919284721674, 6.34620304303927934957474450570, 7.982197569423095370460507509881, 9.031588197240277705946068079454, 9.905074771040248132470443821665, 10.78393469598445035558312701337, 11.83835515331070642536891061116