| L(s) = 1 | + (−6.43 + 26.2i)3-s + 10.3i·5-s + 540.·7-s + (−646. − 337. i)9-s + 1.65e3i·11-s + 1.62e3·13-s + (−271. − 66.7i)15-s − 2.23e3i·17-s + 6.72e3·19-s + (−3.48e3 + 1.41e4i)21-s − 2.12e4i·23-s + 1.55e4·25-s + (1.30e4 − 1.47e4i)27-s − 9.05e3i·29-s + 3.14e4·31-s + ⋯ |
| L(s) = 1 | + (−0.238 + 0.971i)3-s + 0.0828i·5-s + 1.57·7-s + (−0.886 − 0.463i)9-s + 1.24i·11-s + 0.738·13-s + (−0.0804 − 0.0197i)15-s − 0.455i·17-s + 0.980·19-s + (−0.376 + 1.53i)21-s − 1.74i·23-s + 0.993·25-s + (0.661 − 0.750i)27-s − 0.371i·29-s + 1.05·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.415678410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.415678410\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (6.43 - 26.2i)T \) |
| good | 5 | \( 1 - 10.3iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 540.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.65e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 1.62e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 2.23e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 6.72e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 2.12e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 9.05e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 3.14e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 2.17e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 1.32e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 6.78e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 8.98e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 2.03e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 9.03e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.01e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 2.46e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 6.63e4iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 2.59e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 4.51e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 3.22e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 6.68e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 7.77e5T + 8.32e11T^{2} \) |
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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
−11.47800037205614873722005564137, −10.72713375066709198597066645844, −9.793799594440469967524745300247, −8.694783124425304671870864558545, −7.76202943428663638508351204874, −6.32819367309081039332793679540, −4.84151039915164575093411285614, −4.51354280194018616090078074263, −2.76165044686122238463202071234, −1.13924963933242094099112332217,
0.868367814585820893886518510167, 1.69569834997984543160249037459, 3.35094999120052129235904376098, 5.10701114611639317739357137935, 5.89425613481998370052837127275, 7.25562724767244417279358409826, 8.184300440916645957288467692618, 8.809013239985158647826032629885, 10.64790734524952498149791968031, 11.41817463404106751522130993498
