| L(s) = 1 | − 0.444·2-s − 9·3-s − 31.8·4-s + 4.00·6-s − 205.·7-s + 28.3·8-s + 81·9-s + 121·11-s + 286.·12-s + 893.·13-s + 91.5·14-s + 1.00e3·16-s − 836.·17-s − 36.0·18-s + 816.·19-s + 1.85e3·21-s − 53.8·22-s − 3.12e3·23-s − 255.·24-s − 397.·26-s − 729·27-s + 6.54e3·28-s − 7.30e3·29-s + 152.·31-s − 1.35e3·32-s − 1.08e3·33-s + 371.·34-s + ⋯ |
| L(s) = 1 | − 0.0786·2-s − 0.577·3-s − 0.993·4-s + 0.0453·6-s − 1.58·7-s + 0.156·8-s + 0.333·9-s + 0.301·11-s + 0.573·12-s + 1.46·13-s + 0.124·14-s + 0.981·16-s − 0.701·17-s − 0.0262·18-s + 0.518·19-s + 0.916·21-s − 0.0236·22-s − 1.23·23-s − 0.0904·24-s − 0.115·26-s − 0.192·27-s + 1.57·28-s − 1.61·29-s + 0.0284·31-s − 0.233·32-s − 0.174·33-s + 0.0551·34-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
| \(L(3)\) |
\(\approx\) |
\(0.3706826481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3706826481\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 0.444T + 32T^{2} \) |
| 7 | \( 1 + 205.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 893.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 836.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 816.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.12e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.30e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 152.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.73e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.60e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.20e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.18e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.28e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 517.T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.02e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.05e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.64e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.51e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.50e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.61e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.04e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.93e4T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.455709363752049839936242609369, −8.875408602948382725191898553333, −7.84117605668804631841892990170, −6.59871737268540538766697140647, −6.08702521347582668292518482018, −5.12795250419173298099979469161, −3.85078484571128203376809505411, −3.44828314695527494894901524046, −1.58725250934802393415173784389, −0.29640139837403879074750780373,
0.29640139837403879074750780373, 1.58725250934802393415173784389, 3.44828314695527494894901524046, 3.85078484571128203376809505411, 5.12795250419173298099979469161, 6.08702521347582668292518482018, 6.59871737268540538766697140647, 7.84117605668804631841892990170, 8.875408602948382725191898553333, 9.455709363752049839936242609369
