| L(s) = 1 | − 39.8·2-s + 1.07e3·4-s − 625·5-s − 2.40e3·7-s − 2.24e4·8-s + 2.48e4·10-s + 5.04e4·11-s + 3.92e3·13-s + 9.56e4·14-s + 3.42e5·16-s + 5.89e5·17-s + 7.96e5·19-s − 6.71e5·20-s − 2.00e6·22-s + 4.36e5·23-s + 3.90e5·25-s − 1.56e5·26-s − 2.57e6·28-s − 1.20e6·29-s + 3.38e6·31-s − 2.15e6·32-s − 2.34e7·34-s + 1.50e6·35-s + 1.70e7·37-s − 3.17e7·38-s + 1.40e7·40-s − 2.58e7·41-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 2.09·4-s − 0.447·5-s − 0.377·7-s − 1.93·8-s + 0.787·10-s + 1.03·11-s + 0.0380·13-s + 0.665·14-s + 1.30·16-s + 1.71·17-s + 1.40·19-s − 0.938·20-s − 1.82·22-s + 0.325·23-s + 0.200·25-s − 0.0670·26-s − 0.793·28-s − 0.317·29-s + 0.658·31-s − 0.363·32-s − 3.01·34-s + 0.169·35-s + 1.49·37-s − 2.46·38-s + 0.864·40-s − 1.42·41-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
| \(L(5)\) |
\(\approx\) |
\(1.141916296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.141916296\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 39.8T + 512T^{2} \) |
| 11 | \( 1 - 5.04e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 3.92e3T + 1.06e10T^{2} \) |
| 17 | \( 1 - 5.89e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 7.96e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 4.36e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.20e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.38e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.70e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.58e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.44e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.55e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 7.67e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.15e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.54e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.72e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.24e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 8.21e6T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.20e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 9.36e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.94e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.48e8T + 7.60e17T^{2} \) |
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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
−9.777497877140501812242607495740, −9.333069101058743249490228358483, −8.265400021460382645306949699131, −7.50572750572702411865375233654, −6.73892153935664579780617888258, −5.56821154369108649396167391361, −3.78550401374780978299405977893, −2.72079265643569120408344425740, −1.22219382139851094785246223616, −0.74095438456204885821495939586,
0.74095438456204885821495939586, 1.22219382139851094785246223616, 2.72079265643569120408344425740, 3.78550401374780978299405977893, 5.56821154369108649396167391361, 6.73892153935664579780617888258, 7.50572750572702411865375233654, 8.265400021460382645306949699131, 9.333069101058743249490228358483, 9.777497877140501812242607495740
