| L(s) = 1 | − 6.10·2-s + 9·3-s + 5.27·4-s − 41.8·5-s − 54.9·6-s + 208.·7-s + 163.·8-s + 81·9-s + 255.·10-s + 149.·11-s + 47.5·12-s − 952.·13-s − 1.27e3·14-s − 376.·15-s − 1.16e3·16-s + 1.07e3·17-s − 494.·18-s − 486.·19-s − 220.·20-s + 1.87e3·21-s − 910.·22-s + 2.19e3·23-s + 1.46e3·24-s − 1.37e3·25-s + 5.81e3·26-s + 729·27-s + 1.09e3·28-s + ⋯ |
| L(s) = 1 | − 1.07·2-s + 0.577·3-s + 0.164·4-s − 0.747·5-s − 0.623·6-s + 1.60·7-s + 0.901·8-s + 0.333·9-s + 0.807·10-s + 0.371·11-s + 0.0952·12-s − 1.56·13-s − 1.73·14-s − 0.431·15-s − 1.13·16-s + 0.900·17-s − 0.359·18-s − 0.309·19-s − 0.123·20-s + 0.927·21-s − 0.401·22-s + 0.866·23-s + 0.520·24-s − 0.440·25-s + 1.68·26-s + 0.192·27-s + 0.265·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.264824816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.264824816\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 59 | \( 1 + 3.48e3T \) |
| good | 2 | \( 1 + 6.10T + 32T^{2} \) |
| 5 | \( 1 + 41.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 208.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 149.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 952.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.07e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 486.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.19e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.54e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.02e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.16e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.45e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.33e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.58e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 746.T + 4.18e8T^{2} \) |
| 61 | \( 1 - 3.16e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.69e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.70e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.54e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.57e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.64e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.84e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.99e4T + 8.58e9T^{2} \) |
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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
−11.57614158117168556718453558247, −10.66815967795605534695140227173, −9.573055528351024616010617372114, −8.691286995673604855805974113495, −7.68057498949647701418881216670, −7.45704661438709648764973120736, −5.06993732018555113867176737875, −4.11075152170386331032581644027, −2.17107758571453173882624091204, −0.848080708115706036144041214948,
0.848080708115706036144041214948, 2.17107758571453173882624091204, 4.11075152170386331032581644027, 5.06993732018555113867176737875, 7.45704661438709648764973120736, 7.68057498949647701418881216670, 8.691286995673604855805974113495, 9.573055528351024616010617372114, 10.66815967795605534695140227173, 11.57614158117168556718453558247
