| L(s) = 1 | + 3.16·2-s − 9·3-s − 21.9·4-s − 28.4·6-s + 223.·7-s − 170.·8-s + 81·9-s + 121·11-s + 197.·12-s + 999.·13-s + 707.·14-s + 162.·16-s + 2.01e3·17-s + 256.·18-s − 856.·19-s − 2.01e3·21-s + 382.·22-s + 2.37e3·23-s + 1.53e3·24-s + 3.16e3·26-s − 729·27-s − 4.91e3·28-s + 4.71e3·29-s − 9.11e3·31-s + 5.98e3·32-s − 1.08e3·33-s + 6.36e3·34-s + ⋯ |
| L(s) = 1 | + 0.559·2-s − 0.577·3-s − 0.686·4-s − 0.323·6-s + 1.72·7-s − 0.943·8-s + 0.333·9-s + 0.301·11-s + 0.396·12-s + 1.63·13-s + 0.964·14-s + 0.158·16-s + 1.68·17-s + 0.186·18-s − 0.544·19-s − 0.995·21-s + 0.168·22-s + 0.936·23-s + 0.544·24-s + 0.917·26-s − 0.192·27-s − 1.18·28-s + 1.04·29-s − 1.70·31-s + 1.03·32-s − 0.174·33-s + 0.944·34-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
| \(L(3)\) |
\(\approx\) |
\(3.046233112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.046233112\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 3.16T + 32T^{2} \) |
| 7 | \( 1 - 223.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 999.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.01e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 856.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.37e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.71e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.11e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.13e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.19e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.60e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.81e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.09e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.89e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.63e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.31e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.27e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.52e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.18e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.73e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.46e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.36e5T + 8.58e9T^{2} \) |
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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
−9.384897433245037825354952705871, −8.505171956261822806929087825179, −7.966204895532718862783957639022, −6.69012448648957592629044496287, −5.57030606653562485152625107151, −5.17847799397588923346627100367, −4.18437952267859031940193030899, −3.39113453725364784004951592565, −1.57980249301412511509472623355, −0.847923877932123714902892218701,
0.847923877932123714902892218701, 1.57980249301412511509472623355, 3.39113453725364784004951592565, 4.18437952267859031940193030899, 5.17847799397588923346627100367, 5.57030606653562485152625107151, 6.69012448648957592629044496287, 7.966204895532718862783957639022, 8.505171956261822806929087825179, 9.384897433245037825354952705871
