| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.346i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.938 + 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(4.841796054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.841796054\) |
| \(L(4)\) |
|
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
−9.833872298063897101171774056956, −8.740744781312751835364877059849, −8.207849098029515999302880691310, −6.88787219385256243897386334409, −5.90553116108400893033782007990, −5.27282263872852297097249701756, −4.31709372198977865555150010454, −2.60773796961710876103003388812, −1.78578669202561453780762462901, −1.01577833712591617395946116062,
1.32249599655135208306906612893, 1.67622763501594608961873647876, 2.80377178256558846813828153585, 4.57079372498901612826160796031, 5.16639588409500746504397374959, 6.11329670385007100515327786898, 7.02171974854343266066615866115, 8.152154966573516997141442804767, 9.151285011926224413130789996791, 9.645760291742927040502815305800
