| L(s) = 1 | − 24.3·2-s + 124.·3-s + 81.4·4-s − 625·5-s − 3.03e3·6-s + 1.04e4·8-s − 4.14e3·9-s + 1.52e4·10-s − 7.91e4·11-s + 1.01e4·12-s − 1.58e5·13-s − 7.79e4·15-s − 2.97e5·16-s − 4.86e5·17-s + 1.00e5·18-s + 2.09e5·19-s − 5.09e4·20-s + 1.92e6·22-s − 1.31e6·23-s + 1.30e6·24-s + 3.90e5·25-s + 3.86e6·26-s − 2.97e6·27-s − 5.04e6·29-s + 1.89e6·30-s + 6.50e6·31-s + 1.87e6·32-s + ⋯ |
| L(s) = 1 | − 1.07·2-s + 0.888·3-s + 0.159·4-s − 0.447·5-s − 0.956·6-s + 0.905·8-s − 0.210·9-s + 0.481·10-s − 1.63·11-s + 0.141·12-s − 1.54·13-s − 0.397·15-s − 1.13·16-s − 1.41·17-s + 0.226·18-s + 0.368·19-s − 0.0711·20-s + 1.75·22-s − 0.977·23-s + 0.804·24-s + 0.200·25-s + 1.65·26-s − 1.07·27-s − 1.32·29-s + 0.427·30-s + 1.26·31-s + 0.315·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.07487488823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07487488823\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 625T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 24.3T + 512T^{2} \) |
| 3 | \( 1 - 124.T + 1.96e4T^{2} \) |
| 11 | \( 1 + 7.91e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.58e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 4.86e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.09e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.31e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.04e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.50e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.80e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.96e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.68e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.26e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.02e8T + 3.29e15T^{2} \) |
| 59 | \( 1 + 8.22e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 2.00e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.30e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.63e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.12e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.90e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.76e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.55e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.18e9T + 7.60e17T^{2} \) |
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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
−10.20591321999501514505852337993, −9.392150382177570903200942609418, −8.504596384598403676559619641466, −7.82641669949135520808248499292, −7.18012742977431923258669689751, −5.31774944193422545142627735203, −4.26033432915494265359763872974, −2.75224910115996253884463233877, −2.02235831601865576200366073674, −0.13183067036663518930499069917,
0.13183067036663518930499069917, 2.02235831601865576200366073674, 2.75224910115996253884463233877, 4.26033432915494265359763872974, 5.31774944193422545142627735203, 7.18012742977431923258669689751, 7.82641669949135520808248499292, 8.504596384598403676559619641466, 9.392150382177570903200942609418, 10.20591321999501514505852337993
