| L(s) = 1 | − 11.2·3-s − 26.0·5-s − 115.·9-s − 434.·11-s − 737.·13-s + 294.·15-s + 1.41e3·17-s − 1.73e3·19-s − 929.·23-s − 2.44e3·25-s + 4.04e3·27-s − 1.63e3·29-s + 1.87e3·31-s + 4.90e3·33-s − 7.93e3·37-s + 8.32e3·39-s − 6.32e3·41-s − 2.00e4·43-s + 3.01e3·45-s − 1.02e4·47-s − 1.59e4·51-s + 1.19e4·53-s + 1.13e4·55-s + 1.96e4·57-s + 2.11e4·59-s + 350.·61-s + 1.92e4·65-s + ⋯ |
| L(s) = 1 | − 0.724·3-s − 0.466·5-s − 0.475·9-s − 1.08·11-s − 1.20·13-s + 0.337·15-s + 1.18·17-s − 1.10·19-s − 0.366·23-s − 0.782·25-s + 1.06·27-s − 0.360·29-s + 0.349·31-s + 0.784·33-s − 0.952·37-s + 0.876·39-s − 0.587·41-s − 1.65·43-s + 0.221·45-s − 0.677·47-s − 0.858·51-s + 0.585·53-s + 0.505·55-s + 0.799·57-s + 0.792·59-s + 0.0120·61-s + 0.564·65-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
| \(L(3)\) |
\(\approx\) |
\(0.4289929052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4289929052\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 11.2T + 243T^{2} \) |
| 5 | \( 1 + 26.0T + 3.12e3T^{2} \) |
| 11 | \( 1 + 434.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 737.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.41e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.73e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 929.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.63e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.87e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.93e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.32e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.00e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.02e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.19e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.11e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 350.T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.62e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.22e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.37e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.71e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 843.T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.13e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.55e5T + 8.58e9T^{2} \) |
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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
−10.45862620176577003424368812619, −9.882712948974938149425674187039, −8.430024044772238740322825170738, −7.76001258472542175525374914490, −6.66751583571729814555617512331, −5.52195191674568392743785081760, −4.87515491546446130177542264761, −3.44501743663056545533159746039, −2.18503526170290129044270834517, −0.33906178606825960207931512902,
0.33906178606825960207931512902, 2.18503526170290129044270834517, 3.44501743663056545533159746039, 4.87515491546446130177542264761, 5.52195191674568392743785081760, 6.66751583571729814555617512331, 7.76001258472542175525374914490, 8.430024044772238740322825170738, 9.882712948974938149425674187039, 10.45862620176577003424368812619
