| L(s) = 1 | + 4.94·2-s − 5.84·3-s − 7.50·4-s − 28.9·6-s + 1.89·7-s − 195.·8-s − 208.·9-s + 121·11-s + 43.8·12-s + 378.·13-s + 9.37·14-s − 727.·16-s + 1.08e3·17-s − 1.03e3·18-s − 3.11e3·19-s − 11.0·21-s + 598.·22-s + 3.60e3·23-s + 1.14e3·24-s + 1.87e3·26-s + 2.63e3·27-s − 14.2·28-s + 2.83e3·29-s + 3.70e3·31-s + 2.65e3·32-s − 706.·33-s + 5.36e3·34-s + ⋯ |
| L(s) = 1 | + 0.874·2-s − 0.374·3-s − 0.234·4-s − 0.327·6-s + 0.0146·7-s − 1.08·8-s − 0.859·9-s + 0.301·11-s + 0.0878·12-s + 0.620·13-s + 0.0127·14-s − 0.710·16-s + 0.909·17-s − 0.752·18-s − 1.98·19-s − 0.00547·21-s + 0.263·22-s + 1.42·23-s + 0.404·24-s + 0.543·26-s + 0.696·27-s − 0.00342·28-s + 0.625·29-s + 0.692·31-s + 0.458·32-s − 0.112·33-s + 0.795·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.995312026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.995312026\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 4.94T + 32T^{2} \) |
| 3 | \( 1 + 5.84T + 243T^{2} \) |
| 7 | \( 1 - 1.89T + 1.68e4T^{2} \) |
| 13 | \( 1 - 378.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.08e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 3.11e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.60e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.83e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.70e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.86e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.17e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.87e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.91e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.60e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.27e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.18e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.61e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.15e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.73e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.46e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.12e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.35e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.59e4T + 8.58e9T^{2} \) |
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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
−11.27193681993935076416079505058, −10.28342556638891463832192236202, −8.940345983241856991220511206926, −8.360088663413052992808264443493, −6.65385052392895671653667806823, −5.88114925784969777049938147260, −4.92443700682269733864272781576, −3.85592648064945737381001118877, −2.70600240981807079752408937220, −0.72249425136212315651865618370,
0.72249425136212315651865618370, 2.70600240981807079752408937220, 3.85592648064945737381001118877, 4.92443700682269733864272781576, 5.88114925784969777049938147260, 6.65385052392895671653667806823, 8.360088663413052992808264443493, 8.940345983241856991220511206926, 10.28342556638891463832192236202, 11.27193681993935076416079505058
