| L(s) = 1 | + 7.05i·3-s + 140.·5-s − 450.·7-s + 679.·9-s − 746.·11-s + 485. i·13-s + 991. i·15-s − 1.73e3·17-s + (411. − 6.84e3i)19-s − 3.17e3i·21-s + 2.37e3·23-s + 4.14e3·25-s + 9.93e3i·27-s + 2.53e3i·29-s − 2.96e4i·31-s + ⋯ |
| L(s) = 1 | + 0.261i·3-s + 1.12·5-s − 1.31·7-s + 0.931·9-s − 0.561·11-s + 0.220i·13-s + 0.293i·15-s − 0.354·17-s + (0.0600 − 0.998i)19-s − 0.343i·21-s + 0.195·23-s + 0.265·25-s + 0.504i·27-s + 0.104i·29-s − 0.994i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0600 + 0.998i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.0600 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.387938467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.387938467\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-411. + 6.84e3i)T \) |
| good | 3 | \( 1 - 7.05iT - 729T^{2} \) |
| 5 | \( 1 - 140.T + 1.56e4T^{2} \) |
| 7 | \( 1 + 450.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 746.T + 1.77e6T^{2} \) |
| 13 | \( 1 - 485. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 1.73e3T + 2.41e7T^{2} \) |
| 23 | \( 1 - 2.37e3T + 1.48e8T^{2} \) |
| 29 | \( 1 - 2.53e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 2.96e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 3.19e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 2.58e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 2.30e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 5.57e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + 2.22e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 2.74e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 8.50e4T + 5.15e10T^{2} \) |
| 67 | \( 1 - 2.89e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 2.24e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 1.37e4T + 1.51e11T^{2} \) |
| 79 | \( 1 + 4.97e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 1.63e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 1.05e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.12e6iT - 8.32e11T^{2} \) |
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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
−10.12680960247552203362602925113, −9.690438624948923516855940321817, −8.895554926028182600112607642298, −7.30649402330998439772108088266, −6.50187854290202705577524214567, −5.55341150859749268727409048630, −4.34800104048588961167929703299, −3.01837147355183429208546651609, −1.91071542932752147615661394346, −0.32482533358189856444741094463,
1.23546960569352215034663827853, 2.40800252870447376173909329758, 3.60147385611444940171835430542, 5.09960001529243656621380985770, 6.19270057547052762593800326036, 6.81727380050551148367908365298, 8.026465526809120243766707811246, 9.344612710725964925750204703074, 9.970480975566160963224024474002, 10.56343475628646247079976763659
