| L(s) = 1 | − 22.7·2-s + 6.29·4-s + 625·5-s + 2.40e3·7-s + 1.15e4·8-s − 1.42e4·10-s + 2.12e4·11-s + 4.73e4·13-s − 5.46e4·14-s − 2.65e5·16-s − 5.69e5·17-s + 1.08e6·19-s + 3.93e3·20-s − 4.83e5·22-s + 1.34e6·23-s + 3.90e5·25-s − 1.07e6·26-s + 1.51e4·28-s + 6.23e6·29-s − 4.13e5·31-s + 1.45e5·32-s + 1.29e7·34-s + 1.50e6·35-s + 1.28e7·37-s − 2.48e7·38-s + 7.19e6·40-s − 3.90e6·41-s + ⋯ |
| L(s) = 1 | − 1.00·2-s + 0.0122·4-s + 0.447·5-s + 0.377·7-s + 0.993·8-s − 0.449·10-s + 0.437·11-s + 0.460·13-s − 0.380·14-s − 1.01·16-s − 1.65·17-s + 1.91·19-s + 0.00550·20-s − 0.439·22-s + 1.00·23-s + 0.200·25-s − 0.463·26-s + 0.00464·28-s + 1.63·29-s − 0.0804·31-s + 0.0245·32-s + 1.66·34-s + 0.169·35-s + 1.12·37-s − 1.92·38-s + 0.444·40-s − 0.215·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.644331032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.644331032\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 625T \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 2 | \( 1 + 22.7T + 512T^{2} \) |
| 11 | \( 1 - 2.12e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 4.73e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 5.69e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.08e6T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.34e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.23e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.13e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.28e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.90e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.27e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.61e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.10e8T + 3.29e15T^{2} \) |
| 59 | \( 1 - 8.68e6T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.06e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.89e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 9.10e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 9.84e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.41e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.28e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.29e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 3.01e8T + 7.60e17T^{2} \) |
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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
−9.862482236982991655808464322603, −9.147200662194956688007916594442, −8.477489650228658856168890621825, −7.42104388028095596497692580401, −6.50611186426605941081963230154, −5.16121767966419257553177566298, −4.25567832939604624265446161751, −2.72871894271178897455531388988, −1.43211930527215041113664991390, −0.73894454603894110129504733716,
0.73894454603894110129504733716, 1.43211930527215041113664991390, 2.72871894271178897455531388988, 4.25567832939604624265446161751, 5.16121767966419257553177566298, 6.50611186426605941081963230154, 7.42104388028095596497692580401, 8.477489650228658856168890621825, 9.147200662194956688007916594442, 9.862482236982991655808464322603
