| L(s) = 1 | − 12·9-s + 64·11-s + 64·19-s + 80·41-s − 152·49-s − 256·59-s + 90·81-s + 400·89-s − 768·99-s + 1.33e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 584·169-s − 768·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 4/3·9-s + 5.81·11-s + 3.36·19-s + 1.95·41-s − 3.10·49-s − 4.33·59-s + 10/9·81-s + 4.49·89-s − 7.75·99-s + 11.0·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 3.45·169-s − 4.49·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(8.700590711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.700590711\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p T^{2} )^{4} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 76 T^{2} + 3174 T^{4} + 76 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 - 8 T + p^{2} T^{2} )^{8} \) |
| 13 | \( ( 1 + 292 T^{2} + 75366 T^{4} + 292 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 764 T^{2} + 309894 T^{4} - 764 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 16 T + 738 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 + 1636 T^{2} + 1179654 T^{4} + 1636 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 1756 T^{2} + 1539558 T^{4} - 1756 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 - 460 T^{2} - 683610 T^{4} - 460 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 + 4612 T^{2} + 8955366 T^{4} + 4612 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 20 T + 1734 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 43 | \( ( 1 - 6404 T^{2} + 16979814 T^{4} - 6404 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 + 5284 T^{2} + 13593798 T^{4} + 5284 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 + 9436 T^{2} + 38033574 T^{4} + 9436 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + 64 T + 7554 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 61 | \( ( 1 - 11332 T^{2} + 56649510 T^{4} - 11332 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 8900 T^{2} + 53026854 T^{4} - 8900 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 11236 T^{2} + 75307398 T^{4} - 11236 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 10172 T^{2} + 51943878 T^{4} - 10172 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 17548 T^{2} + 151931046 T^{4} - 17548 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 - 22820 T^{2} + 220189542 T^{4} - 22820 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 100 T + 11430 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 - 23420 T^{2} + 308763654 T^{4} - 23420 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.54054929815498144738713041560, −3.53245166145673458358494026631, −3.21159018899480398394511886830, −3.13455609734993997479779468836, −3.08818675309701973478219427254, −2.92168988640578355201725307996, −2.88310892130008498637532689670, −2.84672458428972943285496401596, −2.63342040935333157893365804560, −2.58827892772645941084525144585, −2.21380588424890074672760445386, −1.96258087841890077337940795569, −1.87298695247239732650182061164, −1.80984894778635791180269131987, −1.73165533471923084283143840118, −1.59019475405075706051009479283, −1.42614596073269688207839314089, −1.16187847147120244691358492150, −1.07941816437746383630450164162, −1.07782830106705837282122189406, −1.05798831248922044814100843820, −0.66830002854081748297281756899, −0.64670760634583276977518631817, −0.24647745372705405613877797920, −0.14405874474081469418370575862,
0.14405874474081469418370575862, 0.24647745372705405613877797920, 0.64670760634583276977518631817, 0.66830002854081748297281756899, 1.05798831248922044814100843820, 1.07782830106705837282122189406, 1.07941816437746383630450164162, 1.16187847147120244691358492150, 1.42614596073269688207839314089, 1.59019475405075706051009479283, 1.73165533471923084283143840118, 1.80984894778635791180269131987, 1.87298695247239732650182061164, 1.96258087841890077337940795569, 2.21380588424890074672760445386, 2.58827892772645941084525144585, 2.63342040935333157893365804560, 2.84672458428972943285496401596, 2.88310892130008498637532689670, 2.92168988640578355201725307996, 3.08818675309701973478219427254, 3.13455609734993997479779468836, 3.21159018899480398394511886830, 3.53245166145673458358494026631, 3.54054929815498144738713041560
Plot not available for L-functions of degree greater than 10.
