| L(s) = 1 | − 28.9·3-s − 63.5·5-s + 593.·9-s + 592.·11-s + 433.·13-s + 1.83e3·15-s + 32.2·17-s + 2.71e3·19-s + 3.34e3·23-s + 911.·25-s − 1.01e4·27-s + 8.37e3·29-s + 3.30e3·31-s − 1.71e4·33-s − 812.·37-s − 1.25e4·39-s − 8.71e3·41-s + 9.72e3·43-s − 3.77e4·45-s + 2.02e4·47-s − 931.·51-s − 9.24e3·53-s − 3.76e4·55-s − 7.84e4·57-s + 4.03e3·59-s + 1.08e4·61-s − 2.75e4·65-s + ⋯ |
| L(s) = 1 | − 1.85·3-s − 1.13·5-s + 2.44·9-s + 1.47·11-s + 0.711·13-s + 2.10·15-s + 0.0270·17-s + 1.72·19-s + 1.31·23-s + 0.291·25-s − 2.67·27-s + 1.84·29-s + 0.616·31-s − 2.73·33-s − 0.0976·37-s − 1.31·39-s − 0.809·41-s + 0.801·43-s − 2.77·45-s + 1.33·47-s − 0.0501·51-s − 0.452·53-s − 1.67·55-s − 3.19·57-s + 0.151·59-s + 0.374·61-s − 0.808·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.328582441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.328582441\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 28.9T + 243T^{2} \) |
| 5 | \( 1 + 63.5T + 3.12e3T^{2} \) |
| 11 | \( 1 - 592.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 433.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 32.2T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.71e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.34e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.37e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.30e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 812.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 8.71e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.72e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.02e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.24e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.03e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.08e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.27e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.22e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.07e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.98e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.64e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.48e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.96e4T + 8.58e9T^{2} \) |
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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
−9.729453785915402968593267172875, −8.717919353160189195309478059009, −7.48812976150430726643429608172, −6.81974524631339871573276629809, −6.09903446309787851583992669837, −5.06454733747955108408427717092, −4.29775340260411539503580771301, −3.39947993968634748494103914474, −1.12560671020745800498075026829, −0.77262241535927582462425627279,
0.77262241535927582462425627279, 1.12560671020745800498075026829, 3.39947993968634748494103914474, 4.29775340260411539503580771301, 5.06454733747955108408427717092, 6.09903446309787851583992669837, 6.81974524631339871573276629809, 7.48812976150430726643429608172, 8.717919353160189195309478059009, 9.729453785915402968593267172875
