| L(s) = 1 | + 11.1·2-s + 9·3-s + 91.3·4-s + 99.9·6-s + 76.1·7-s + 659.·8-s + 81·9-s − 121·11-s + 822.·12-s − 470.·13-s + 845.·14-s + 4.39e3·16-s − 494.·17-s + 899.·18-s − 44.8·19-s + 685.·21-s − 1.34e3·22-s + 4.24e3·23-s + 5.93e3·24-s − 5.22e3·26-s + 729·27-s + 6.95e3·28-s − 786.·29-s + 5.59e3·31-s + 2.77e4·32-s − 1.08e3·33-s − 5.49e3·34-s + ⋯ |
| L(s) = 1 | + 1.96·2-s + 0.577·3-s + 2.85·4-s + 1.13·6-s + 0.587·7-s + 3.64·8-s + 0.333·9-s − 0.301·11-s + 1.64·12-s − 0.772·13-s + 1.15·14-s + 4.29·16-s − 0.415·17-s + 0.654·18-s − 0.0284·19-s + 0.339·21-s − 0.591·22-s + 1.67·23-s + 2.10·24-s − 1.51·26-s + 0.192·27-s + 1.67·28-s − 0.173·29-s + 1.04·31-s + 4.79·32-s − 0.174·33-s − 0.815·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(12.66243593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.66243593\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 - 11.1T + 32T^{2} \) |
| 7 | \( 1 - 76.1T + 1.68e4T^{2} \) |
| 13 | \( 1 + 470.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 494.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 44.8T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.24e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 786.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.45e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.70e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.42e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.88e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.44e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.79e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.46e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.72e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.95e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.69e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.22e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.43e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.13e4T + 8.58e9T^{2} \) |
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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
−9.653266877033653703682099675431, −8.295473911324427228755389593683, −7.42551551885389354956027393080, −6.78399979653963614390096146149, −5.69153434766996811388730571405, −4.80542523304399719877910140329, −4.27297842636112378332065039768, −3.02541371265508746510821935859, −2.46227655960143822983296687420, −1.31465515438004407487149419357,
1.31465515438004407487149419357, 2.46227655960143822983296687420, 3.02541371265508746510821935859, 4.27297842636112378332065039768, 4.80542523304399719877910140329, 5.69153434766996811388730571405, 6.78399979653963614390096146149, 7.42551551885389354956027393080, 8.295473911324427228755389593683, 9.653266877033653703682099675431
