| L(s) = 1 | + (354. + 613. i)5-s + (−728. − 420. i)7-s + (1.00e4 + 5.80e3i)11-s + (−190. − 329. i)13-s − 1.31e3·17-s − 8.35e4i·19-s + (4.80e4 − 2.77e4i)23-s + (−5.55e4 + 9.62e4i)25-s + (3.76e5 − 6.52e5i)29-s + (−1.29e4 + 7.50e3i)31-s − 5.96e5i·35-s − 8.61e5·37-s + (−1.07e6 − 1.85e6i)41-s + (3.22e6 + 1.86e6i)43-s + (−2.67e5 − 1.54e5i)47-s + ⋯ |
| L(s) = 1 | + (0.566 + 0.981i)5-s + (−0.303 − 0.175i)7-s + (0.687 + 0.396i)11-s + (−0.00665 − 0.0115i)13-s − 0.0157·17-s − 0.641i·19-s + (0.171 − 0.0990i)23-s + (−0.142 + 0.246i)25-s + (0.532 − 0.922i)29-s + (−0.0140 + 0.00812i)31-s − 0.397i·35-s − 0.459·37-s + (−0.379 − 0.656i)41-s + (0.944 + 0.545i)43-s + (−0.0548 − 0.0316i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.370i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.928 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.370900139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.370900139\) |
| \(L(5)\) |
|
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 | \( 1 + (-354. - 613. i)T + (-1.95e5 + 3.38e5i)T^{2} \) |
| 7 | \( 1 + (728. + 420. i)T + (2.88e6 + 4.99e6i)T^{2} \) |
| 11 | \( 1 + (-1.00e4 - 5.80e3i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + (190. + 329. i)T + (-4.07e8 + 7.06e8i)T^{2} \) |
| 17 | \( 1 + 1.31e3T + 6.97e9T^{2} \) |
| 19 | \( 1 + 8.35e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 + (-4.80e4 + 2.77e4i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-3.76e5 + 6.52e5i)T + (-2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 + (1.29e4 - 7.50e3i)T + (4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + 8.61e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + (1.07e6 + 1.85e6i)T + (-3.99e12 + 6.91e12i)T^{2} \) |
| 43 | \( 1 + (-3.22e6 - 1.86e6i)T + (5.84e12 + 1.01e13i)T^{2} \) |
| 47 | \( 1 + (2.67e5 + 1.54e5i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + 7.04e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + (-8.17e6 + 4.72e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-1.06e7 + 1.84e7i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (4.31e6 - 2.48e6i)T + (2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 4.06e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 1.84e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + (-4.75e7 - 2.74e7i)T + (7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (3.76e7 + 2.17e7i)T + (1.12e15 + 1.95e15i)T^{2} \) |
| 89 | \( 1 - 1.13e8T + 3.93e15T^{2} \) |
| 97 | \( 1 + (5.49e7 - 9.51e7i)T + (-3.91e15 - 6.78e15i)T^{2} \) |
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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
−9.823914962666694160540377754637, −9.052376344186497370214643740788, −7.82084179774101985721761784139, −6.74099730265835961427964534896, −6.35909681151716030742658879254, −5.04888900110923519572185084880, −3.86658915157961024426806424175, −2.82246528694698547773669786439, −1.88828090856322157583122842254, −0.51876911652583395505262998757,
0.887683997827130396270719513572, 1.65814534251483829245724721476, 3.02522968673967080700654355402, 4.17446382042248711148836657368, 5.24540362561698113561927049151, 6.01994024158608772983132410038, 7.02500979868054601523359536473, 8.306606881201092789417669147166, 8.994497161428141260462443441082, 9.692840217971470395574286334447
