| L(s) = 1 | − 1.96·2-s + 27·3-s − 124.·4-s + 495.·5-s − 53.0·6-s + 824.·7-s + 495.·8-s + 729·9-s − 973.·10-s − 5.18e3·11-s − 3.35e3·12-s − 1.24e3·13-s − 1.61e3·14-s + 1.33e4·15-s + 1.49e4·16-s + 1.11e4·17-s − 1.43e3·18-s + 1.09e4·19-s − 6.15e4·20-s + 2.22e4·21-s + 1.01e4·22-s + 1.21e4·23-s + 1.33e4·24-s + 1.67e5·25-s + 2.45e3·26-s + 1.96e4·27-s − 1.02e5·28-s + ⋯ |
| L(s) = 1 | − 0.173·2-s + 0.577·3-s − 0.969·4-s + 1.77·5-s − 0.100·6-s + 0.908·7-s + 0.342·8-s + 0.333·9-s − 0.307·10-s − 1.17·11-s − 0.559·12-s − 0.157·13-s − 0.157·14-s + 1.02·15-s + 0.910·16-s + 0.552·17-s − 0.0578·18-s + 0.366·19-s − 1.71·20-s + 0.524·21-s + 0.203·22-s + 0.208·23-s + 0.197·24-s + 2.14·25-s + 0.0273·26-s + 0.192·27-s − 0.881·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.571087331\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.571087331\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 23 | \( 1 - 1.21e4T \) |
| good | 2 | \( 1 + 1.96T + 128T^{2} \) |
| 5 | \( 1 - 495.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 824.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.18e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.24e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.11e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.09e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 2.49e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.80e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 9.33e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.88e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.93e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.25e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.37e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 7.08e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.95e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.42e4T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.91e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.89e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.32e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.49e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.14e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.27e7T + 8.07e13T^{2} \) |
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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
−13.66791282896449612498950787234, −12.57966354543254660068291688740, −10.54272231473674269236204410874, −9.801365082573233116375761422993, −8.797474815183569246756570403647, −7.70115225138059634800859028782, −5.70685864930418841370542725482, −4.76438724005991075317311137895, −2.62916581769461796212104820489, −1.23185850544267533210518505911,
1.23185850544267533210518505911, 2.62916581769461796212104820489, 4.76438724005991075317311137895, 5.70685864930418841370542725482, 7.70115225138059634800859028782, 8.797474815183569246756570403647, 9.801365082573233116375761422993, 10.54272231473674269236204410874, 12.57966354543254660068291688740, 13.66791282896449612498950787234
