| L(s) = 1 | + 2.30·3-s − 81.0·5-s − 237.·9-s − 732.·11-s − 919.·13-s − 186.·15-s − 1.72e3·17-s + 2.44e3·19-s + 2.42e3·23-s + 3.45e3·25-s − 1.10e3·27-s + 1.01e3·29-s − 6.12e3·31-s − 1.68e3·33-s − 1.23e3·37-s − 2.11e3·39-s − 1.53e4·41-s + 1.29e4·43-s + 1.92e4·45-s + 6.55e3·47-s − 3.98e3·51-s − 3.15e4·53-s + 5.93e4·55-s + 5.62e3·57-s + 2.18e4·59-s − 2.70e4·61-s + 7.45e4·65-s + ⋯ |
| L(s) = 1 | + 0.147·3-s − 1.45·5-s − 0.978·9-s − 1.82·11-s − 1.50·13-s − 0.214·15-s − 1.45·17-s + 1.55·19-s + 0.955·23-s + 1.10·25-s − 0.292·27-s + 0.223·29-s − 1.14·31-s − 0.269·33-s − 0.148·37-s − 0.222·39-s − 1.42·41-s + 1.07·43-s + 1.41·45-s + 0.433·47-s − 0.214·51-s − 1.54·53-s + 2.64·55-s + 0.229·57-s + 0.818·59-s − 0.929·61-s + 2.18·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2661006678\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2661006678\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 2.30T + 243T^{2} \) |
| 5 | \( 1 + 81.0T + 3.12e3T^{2} \) |
| 11 | \( 1 + 732.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 919.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.72e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.44e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.42e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.01e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.12e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.23e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.53e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.29e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.55e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.15e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.18e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.70e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 8.26e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.71e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 9.28e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.91e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.08e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.78e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.66e4T + 8.58e9T^{2} \) |
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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
−10.74122617749097021552337516478, −9.489260389455249082114713281785, −8.477721055843037506305986830188, −7.65926072189267078335688750361, −7.12155520366449994887234434094, −5.37469547776589798879834129774, −4.72415042423868416203707774707, −3.26599083783188950275749377197, −2.49942470610534284340073733498, −0.25085236517063360918497978712,
0.25085236517063360918497978712, 2.49942470610534284340073733498, 3.26599083783188950275749377197, 4.72415042423868416203707774707, 5.37469547776589798879834129774, 7.12155520366449994887234434094, 7.65926072189267078335688750361, 8.477721055843037506305986830188, 9.489260389455249082114713281785, 10.74122617749097021552337516478
