| L(s) = 1 | + 5.74·2-s − 214.·3-s − 478.·4-s + 2.39e3·5-s − 1.23e3·6-s − 2.77e3·7-s − 5.69e3·8-s + 2.62e4·9-s + 1.37e4·10-s + 5.44e4·11-s + 1.02e5·12-s + 1.87e5·13-s − 1.59e4·14-s − 5.12e5·15-s + 2.12e5·16-s + 1.50e5·18-s + 7.56e5·19-s − 1.14e6·20-s + 5.93e5·21-s + 3.12e5·22-s + 6.48e5·23-s + 1.21e6·24-s + 3.77e6·25-s + 1.07e6·26-s − 1.39e6·27-s + 1.32e6·28-s + 1.04e6·29-s + ⋯ |
| L(s) = 1 | + 0.253·2-s − 1.52·3-s − 0.935·4-s + 1.71·5-s − 0.387·6-s − 0.436·7-s − 0.491·8-s + 1.33·9-s + 0.434·10-s + 1.12·11-s + 1.42·12-s + 1.82·13-s − 0.110·14-s − 2.61·15-s + 0.810·16-s + 0.337·18-s + 1.33·19-s − 1.60·20-s + 0.666·21-s + 0.284·22-s + 0.482·23-s + 0.750·24-s + 1.93·25-s + 0.463·26-s − 0.505·27-s + 0.408·28-s + 0.275·29-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
| \(L(5)\) |
\(\approx\) |
\(2.273122885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.273122885\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 5.74T + 512T^{2} \) |
| 3 | \( 1 + 214.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.39e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 2.77e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.44e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.87e5T + 1.06e10T^{2} \) |
| 19 | \( 1 - 7.56e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 6.48e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.04e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.10e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 8.95e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.34e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.81e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.47e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.72e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 9.17e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.39e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.95e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 5.02e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.39e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.15e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.34e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.29e5T + 3.50e17T^{2} \) |
| 97 | \( 1 + 8.51e8T + 7.60e17T^{2} \) |
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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
−10.17254058630975443615362018728, −9.486747655939324068237840802365, −8.690060570133501307632593716772, −6.68736253699972980672453475170, −6.05622042398080092694562458024, −5.55226549640271978548957176103, −4.54007497325277304749596927008, −3.26141564563372876009046588544, −1.29854803378919387516797164951, −0.866510849237946098742316376372,
0.866510849237946098742316376372, 1.29854803378919387516797164951, 3.26141564563372876009046588544, 4.54007497325277304749596927008, 5.55226549640271978548957176103, 6.05622042398080092694562458024, 6.68736253699972980672453475170, 8.690060570133501307632593716772, 9.486747655939324068237840802365, 10.17254058630975443615362018728
