| L(s) = 1 | + 6.27·5-s + 7·7-s − 54.3·11-s + 40.9·13-s + 25.0·17-s + 23·19-s + 139.·23-s − 85.6·25-s + 73.1·29-s − 57.7·31-s + 43.9·35-s + 277.·37-s + 346.·41-s + 364.·43-s − 599.·47-s + 49·49-s − 91.7·53-s − 340.·55-s − 771.·59-s + 373.·61-s + 256.·65-s + 718.·67-s + 248.·71-s + 376.·73-s − 380.·77-s + 305.·79-s + 210.·83-s + ⋯ |
| L(s) = 1 | + 0.561·5-s + 0.377·7-s − 1.48·11-s + 0.872·13-s + 0.357·17-s + 0.277·19-s + 1.26·23-s − 0.685·25-s + 0.468·29-s − 0.334·31-s + 0.212·35-s + 1.23·37-s + 1.32·41-s + 1.29·43-s − 1.86·47-s + 0.142·49-s − 0.237·53-s − 0.835·55-s − 1.70·59-s + 0.784·61-s + 0.489·65-s + 1.30·67-s + 0.415·71-s + 0.604·73-s − 0.562·77-s + 0.435·79-s + 0.278·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.331200703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.331200703\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 - 6.27T + 125T^{2} \) |
| 11 | \( 1 + 54.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 40.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 25.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 23T + 6.85e3T^{2} \) |
| 23 | \( 1 - 139.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 73.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 57.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 277.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 346.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 364.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 599.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 91.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + 771.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 373.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 718.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 248.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 376.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 305.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 210.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 133.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.04e3T + 9.12e5T^{2} \) |
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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
−9.942724587957326844586119380589, −9.174080002882724426489967521654, −8.121950110620615552268941852388, −7.54236531803772814737829718544, −6.27917842640218929015572815074, −5.50171240029130999441704167347, −4.65417633400599595091390778243, −3.26866110878601584404484879085, −2.22749807532352957297683744708, −0.885551592954025603315412664576,
0.885551592954025603315412664576, 2.22749807532352957297683744708, 3.26866110878601584404484879085, 4.65417633400599595091390778243, 5.50171240029130999441704167347, 6.27917842640218929015572815074, 7.54236531803772814737829718544, 8.121950110620615552268941852388, 9.174080002882724426489967521654, 9.942724587957326844586119380589
