| L(s) = 1 | − 11.1·2-s + 93.0·4-s − 682.·8-s + 4.64e3·16-s − 648.·17-s + 2.16e3·19-s + 1.38e3·23-s − 8.15e3·31-s − 3.01e4·32-s + 7.25e3·34-s − 2.41e4·38-s − 1.55e4·46-s − 1.21e4·47-s − 1.68e4·49-s + 4.08e4·53-s + 3.48e4·61-s + 9.11e4·62-s + 1.88e5·64-s − 6.03e4·68-s + 2.01e5·76-s − 7.00e4·79-s + 7.23e4·83-s + 1.28e5·92-s + 1.35e5·94-s + 1.87e5·98-s − 4.57e5·106-s + 2.06e5·107-s + ⋯ |
| L(s) = 1 | − 1.97·2-s + 2.90·4-s − 3.76·8-s + 4.54·16-s − 0.544·17-s + 1.37·19-s + 0.546·23-s − 1.52·31-s − 5.20·32-s + 1.07·34-s − 2.71·38-s − 1.08·46-s − 0.800·47-s − 49-s + 1.99·53-s + 1.19·61-s + 3.01·62-s + 5.74·64-s − 1.58·68-s + 3.99·76-s − 1.26·79-s + 1.15·83-s + 1.58·92-s + 1.58·94-s + 1.97·98-s − 3.95·106-s + 1.74·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7104013056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7104013056\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 11.1T + 32T^{2} \) |
| 7 | \( 1 + 1.68e4T^{2} \) |
| 11 | \( 1 + 1.61e5T^{2} \) |
| 13 | \( 1 + 3.71e5T^{2} \) |
| 17 | \( 1 + 648.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.16e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.38e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.21e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.08e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.48e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.00e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.23e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.58e9T^{2} \) |
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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
−11.15522096013921566452219619234, −10.17241200259441527062338536323, −9.361708172359028214850521729711, −8.592386283982906586339933797149, −7.53949007987571013427857934668, −6.82487559661924134207276735357, −5.56104889237121274645102220485, −3.23871386165419659259230616069, −1.92055001703104033560907114967, −0.67438472766258768548258785426,
0.67438472766258768548258785426, 1.92055001703104033560907114967, 3.23871386165419659259230616069, 5.56104889237121274645102220485, 6.82487559661924134207276735357, 7.53949007987571013427857934668, 8.592386283982906586339933797149, 9.361708172359028214850521729711, 10.17241200259441527062338536323, 11.15522096013921566452219619234
