| L(s) = 1 | + 6.22·3-s − 9.40·5-s − 204.·9-s − 765.·11-s − 732.·13-s − 58.5·15-s − 482.·17-s − 2.61e3·19-s + 535.·23-s − 3.03e3·25-s − 2.78e3·27-s + 3.94e3·29-s + 3.66e3·31-s − 4.76e3·33-s + 1.26e4·37-s − 4.55e3·39-s − 4.81e3·41-s + 4.93e3·43-s + 1.92e3·45-s + 1.74e4·47-s − 3.00e3·51-s − 4.65e3·53-s + 7.19e3·55-s − 1.62e4·57-s + 3.25e4·59-s + 4.34e4·61-s + 6.88e3·65-s + ⋯ |
| L(s) = 1 | + 0.399·3-s − 0.168·5-s − 0.840·9-s − 1.90·11-s − 1.20·13-s − 0.0671·15-s − 0.404·17-s − 1.66·19-s + 0.211·23-s − 0.971·25-s − 0.735·27-s + 0.870·29-s + 0.684·31-s − 0.761·33-s + 1.51·37-s − 0.479·39-s − 0.447·41-s + 0.407·43-s + 0.141·45-s + 1.14·47-s − 0.161·51-s − 0.227·53-s + 0.320·55-s − 0.664·57-s + 1.21·59-s + 1.49·61-s + 0.202·65-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
| \(L(3)\) |
\(\approx\) |
\(0.7665246930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7665246930\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 6.22T + 243T^{2} \) |
| 5 | \( 1 + 9.40T + 3.12e3T^{2} \) |
| 11 | \( 1 + 765.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 732.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 482.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.61e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 535.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.94e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.66e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.26e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.81e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.93e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.74e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.65e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.25e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.34e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.20e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.58e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.65e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.54e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.94e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.08e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.45e4T + 8.58e9T^{2} \) |
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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
−9.593046437150003723868705645243, −8.399942478408922975059854617728, −8.077859851453409748313421062973, −7.08904460387927255966161022560, −5.94362643329376383514914231711, −5.07031826134273222792353095226, −4.13435751225780366027676776567, −2.55975208400537514982384733313, −2.46349576495865863832134408411, −0.36007630621266519854968049680,
0.36007630621266519854968049680, 2.46349576495865863832134408411, 2.55975208400537514982384733313, 4.13435751225780366027676776567, 5.07031826134273222792353095226, 5.94362643329376383514914231711, 7.08904460387927255966161022560, 8.077859851453409748313421062973, 8.399942478408922975059854617728, 9.593046437150003723868705645243
