| L(s) = 1 | − 33.0·2-s − 199.·3-s + 580.·4-s + 108.·5-s + 6.60e3·6-s − 1.47e3·7-s − 2.26e3·8-s + 2.02e4·9-s − 3.59e3·10-s − 8.21e4·11-s − 1.15e5·12-s + 2.36e4·13-s + 4.88e4·14-s − 2.17e4·15-s − 2.22e5·16-s − 6.68e5·18-s − 5.39e5·19-s + 6.31e4·20-s + 2.95e5·21-s + 2.71e6·22-s + 9.60e5·23-s + 4.52e5·24-s − 1.94e6·25-s − 7.80e5·26-s − 1.09e5·27-s − 8.58e5·28-s − 6.36e6·29-s + ⋯ |
| L(s) = 1 | − 1.46·2-s − 1.42·3-s + 1.13·4-s + 0.0778·5-s + 2.08·6-s − 0.232·7-s − 0.195·8-s + 1.02·9-s − 0.113·10-s − 1.69·11-s − 1.61·12-s + 0.229·13-s + 0.339·14-s − 0.110·15-s − 0.848·16-s − 1.50·18-s − 0.948·19-s + 0.0882·20-s + 0.331·21-s + 2.47·22-s + 0.715·23-s + 0.278·24-s − 0.993·25-s − 0.335·26-s − 0.0397·27-s − 0.263·28-s − 1.67·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.01553818422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01553818422\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 33.0T + 512T^{2} \) |
| 3 | \( 1 + 199.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 108.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.47e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 8.21e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 2.36e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 5.39e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 9.60e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.36e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.33e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.03e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.70e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.16e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.24e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.44e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 8.66e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 2.76e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.41e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.01e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.67e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.40e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 7.40e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.04e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 7.98e8T + 7.60e17T^{2} \) |
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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
−10.28581875038444008027802035347, −9.511339177296186892666883256835, −8.281670357638274178831908142049, −7.51532056829559532563813174476, −6.43954501312183655052885911918, −5.55880591800154840045777077616, −4.49264505391001025494296822499, −2.58567326618808650533658633842, −1.33481268623579893412450493587, −0.080135065962744128088013106327,
0.080135065962744128088013106327, 1.33481268623579893412450493587, 2.58567326618808650533658633842, 4.49264505391001025494296822499, 5.55880591800154840045777077616, 6.43954501312183655052885911918, 7.51532056829559532563813174476, 8.281670357638274178831908142049, 9.511339177296186892666883256835, 10.28581875038444008027802035347
