| L(s) = 1 | + 238.·5-s − 590.·7-s − 646.·11-s + 1.79e3i·13-s − 4.42e3i·17-s + 169. i·19-s − 1.56e4i·23-s + 4.13e4·25-s + 1.45e4·29-s − 8.10e3·31-s − 1.40e5·35-s + 8.13e4i·37-s + 3.08e4i·41-s + 9.23e4i·43-s − 1.42e5i·47-s + ⋯ |
| L(s) = 1 | + 1.91·5-s − 1.72·7-s − 0.485·11-s + 0.817i·13-s − 0.899i·17-s + 0.0247i·19-s − 1.28i·23-s + 2.64·25-s + 0.595·29-s − 0.272·31-s − 3.28·35-s + 1.60i·37-s + 0.448i·41-s + 1.16i·43-s − 1.37i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.346 - 0.938i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.346 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.411047108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.411047108\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 238.T + 1.56e4T^{2} \) |
| 7 | \( 1 + 590.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 646.T + 1.77e6T^{2} \) |
| 13 | \( 1 - 1.79e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 4.42e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 169. iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 1.56e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 1.45e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 8.10e3T + 8.87e8T^{2} \) |
| 37 | \( 1 - 8.13e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 3.08e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 9.23e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.42e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.83e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 1.83e4T + 4.21e10T^{2} \) |
| 61 | \( 1 - 2.11e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 4.78e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 5.82e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.51e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 4.44e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 8.70e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 5.72e4iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.53e6T + 8.32e11T^{2} \) |
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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
−9.931961573984123427296986354311, −9.411789033592030498321653377370, −8.597990773286326401176116224284, −6.86690781377579110922288419233, −6.50533270001674586974389040139, −5.66236156541514237978923277164, −4.62392929777493039025485559988, −2.98651468818993087332205734501, −2.44896928418539065603732238238, −1.11161731543816563502850578284,
0.27089779897146296659035464753, 1.64567840359219660234933686827, 2.68416729846531882190582594436, 3.51772014500970301520262915655, 5.23193344786366553332904177710, 5.95236603793337746839638255498, 6.45634029769518653601893759465, 7.61386613712457342227549552302, 9.049645458617077836638411402760, 9.495259070536270960315843398337
