| L(s) = 1 | − 15.5·3-s + 195. i·5-s − 277. i·7-s + 243·9-s + 1.75e3·11-s + 1.24e3i·13-s − 3.04e3i·15-s + 6.88e3·17-s − 4.40e3·19-s + 4.32e3i·21-s − 1.27e4i·23-s − 2.24e4·25-s − 3.78e3·27-s + 8.27e3i·29-s − 4.39e4i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.56i·5-s − 0.809i·7-s + 0.333·9-s + 1.31·11-s + 0.567i·13-s − 0.901i·15-s + 1.40·17-s − 0.641·19-s + 0.467i·21-s − 1.04i·23-s − 1.43·25-s − 0.192·27-s + 0.339i·29-s − 1.47i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.870758573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.870758573\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 15.5T \) |
| good | 5 | \( 1 - 195. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 277. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 1.75e3T + 1.77e6T^{2} \) |
| 13 | \( 1 - 1.24e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 6.88e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 4.40e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 1.27e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 8.27e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 4.39e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 1.21e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 5.47e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 4.54e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.52e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 2.72e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 2.13e5T + 4.21e10T^{2} \) |
| 61 | \( 1 - 8.39e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 3.73e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 6.67e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.99e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 4.35e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 2.46e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 8.09e4T + 4.96e11T^{2} \) |
| 97 | \( 1 - 8.77e5T + 8.32e11T^{2} \) |
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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
−10.26305615288823704583091602596, −9.885046969274707990997406313003, −8.390674932305456644838986304907, −7.00982696633279584563316894571, −6.82931806652982721357836565350, −5.76812288211621321238653464470, −4.19748279194222008486591207253, −3.47409445591221360336973023396, −2.00621709753315856941753238121, −0.61392696382008788483802281054,
0.879527115111115201654765081076, 1.59692100893727783528057529757, 3.46186948844699706787561588811, 4.68447391989761039571903126543, 5.46822332030646707476382502232, 6.24210501707656838724479573869, 7.66217989251477347519927291140, 8.674434112301796818025155576157, 9.274812802001766004273114356865, 10.21650479927784803782423586658
