| L(s) = 1 | + 500·5-s − 729·9-s + 7.98e3·11-s − 5.06e4·19-s + 1.71e5·25-s + 3.05e5·29-s + 2.47e5·31-s + 7.93e5·41-s − 3.64e5·45-s + 1.12e6·49-s + 3.99e6·55-s − 6.04e5·59-s − 5.66e6·61-s + 2.01e6·71-s + 1.50e7·79-s + 5.31e5·81-s + 1.53e7·89-s − 2.53e7·95-s − 5.82e6·99-s + 2.25e7·101-s − 2.34e7·109-s + 8.88e6·121-s + 4.68e7·125-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1/3·9-s + 1.80·11-s − 1.69·19-s + 11/5·25-s + 2.32·29-s + 1.49·31-s + 1.79·41-s − 0.596·45-s + 1.36·49-s + 3.23·55-s − 0.383·59-s − 3.19·61-s + 0.668·71-s + 3.43·79-s + 1/9·81-s + 2.30·89-s − 3.02·95-s − 0.603·99-s + 2.18·101-s − 1.73·109-s + 0.455·121-s + 2.14·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(7.226020524\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.226020524\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{6} T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 p^{3} T + p^{7} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 1125802 T^{2} + p^{14} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3994 T + p^{7} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 116316134 T^{2} + p^{14} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 397058622 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 25320 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2367161790 T^{2} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 152664 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 123776 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 75696805270 T^{2} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 396530 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 347519328310 T^{2} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 984199174302 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 813245470198 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 302354 T + p^{7} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2830198 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 1875692967338 T^{2} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 1007580 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 16312522745698 T^{2} + p^{14} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 7517832 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 26186045040870 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 7650250 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 60474559225090 T^{2} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.73746359499404746358741179629, −10.64090937347092432643641152675, −10.31103248164198665031161062140, −9.438276081438769897736400451286, −9.232636078728057821413161442035, −8.955306405862500210268045754562, −8.291925366307456891326323554897, −7.77770274961447282530907066784, −6.72544077070788453605855867810, −6.48493709322120825277649359293, −6.21421099105344491418977196364, −5.78190968975810680318611012941, −4.68860291764205134105060784412, −4.64830810083126673568302267338, −3.78405352949417760406161617669, −2.90024640634130903153820627753, −2.38837176470579473461195926671, −1.84840200787781451134909371757, −1.05798392935179322955006178721, −0.73131850340374788338587297938,
0.73131850340374788338587297938, 1.05798392935179322955006178721, 1.84840200787781451134909371757, 2.38837176470579473461195926671, 2.90024640634130903153820627753, 3.78405352949417760406161617669, 4.64830810083126673568302267338, 4.68860291764205134105060784412, 5.78190968975810680318611012941, 6.21421099105344491418977196364, 6.48493709322120825277649359293, 6.72544077070788453605855867810, 7.77770274961447282530907066784, 8.291925366307456891326323554897, 8.955306405862500210268045754562, 9.232636078728057821413161442035, 9.438276081438769897736400451286, 10.31103248164198665031161062140, 10.64090937347092432643641152675, 10.73746359499404746358741179629
