| L(s) = 1 | − 0.790·2-s + 9·3-s − 31.3·4-s − 104.·5-s − 7.11·6-s + 50.1·8-s + 81·9-s + 82.3·10-s − 497.·11-s − 282.·12-s − 206.·13-s − 937.·15-s + 964.·16-s + 63.1·17-s − 64.0·18-s + 1.32e3·19-s + 3.26e3·20-s + 393.·22-s − 194.·23-s + 451.·24-s + 7.73e3·25-s + 163.·26-s + 729·27-s + 4.32e3·29-s + 741.·30-s − 7.52e3·31-s − 2.36e3·32-s + ⋯ |
| L(s) = 1 | − 0.139·2-s + 0.577·3-s − 0.980·4-s − 1.86·5-s − 0.0807·6-s + 0.276·8-s + 0.333·9-s + 0.260·10-s − 1.24·11-s − 0.566·12-s − 0.338·13-s − 1.07·15-s + 0.941·16-s + 0.0530·17-s − 0.0465·18-s + 0.841·19-s + 1.82·20-s + 0.173·22-s − 0.0766·23-s + 0.159·24-s + 2.47·25-s + 0.0473·26-s + 0.192·27-s + 0.954·29-s + 0.150·30-s − 1.40·31-s − 0.408·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7927660807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7927660807\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 0.790T + 32T^{2} \) |
| 5 | \( 1 + 104.T + 3.12e3T^{2} \) |
| 11 | \( 1 + 497.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 206.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 63.1T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.32e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 194.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.52e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.03e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.18e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.96e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 4.38e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.77e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.50e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.06e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.32e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.88e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.13e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.94e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.02e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.12e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.03e4T + 8.58e9T^{2} \) |
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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
−12.30392046395936098176315055148, −11.12917455502319321314929667161, −10.00236002354133476817283682335, −8.789375788407874978613129653287, −7.943688682887861900176996803465, −7.38926743240038113432561634796, −5.12989815279090033022644430290, −4.12436066914436054174929921631, −3.05883832929047855325040438754, −0.57607934224820078751726042847,
0.57607934224820078751726042847, 3.05883832929047855325040438754, 4.12436066914436054174929921631, 5.12989815279090033022644430290, 7.38926743240038113432561634796, 7.943688682887861900176996803465, 8.789375788407874978613129653287, 10.00236002354133476817283682335, 11.12917455502319321314929667161, 12.30392046395936098176315055148
