| L(s) = 1 | − 729·9-s + 6.31e3·11-s − 9.14e4·19-s − 7.72e4·29-s + 3.84e5·31-s + 1.72e5·41-s + 9.54e5·49-s − 3.97e5·59-s + 2.41e6·61-s − 9.87e6·71-s − 1.31e7·79-s + 5.31e5·81-s − 1.10e7·89-s − 4.60e6·99-s − 2.19e7·101-s − 4.19e7·109-s − 9.09e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6.63e7·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 1.42·11-s − 3.05·19-s − 0.588·29-s + 2.31·31-s + 0.389·41-s + 1.15·49-s − 0.252·59-s + 1.36·61-s − 3.27·71-s − 2.99·79-s + 1/9·81-s − 1.66·89-s − 0.476·99-s − 2.11·101-s − 3.09·109-s − 0.466·121-s + 1.05·169-s + 1.01·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.5269124962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5269124962\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 954862 T^{2} + p^{14} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3156 T + p^{7} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 66360934 T^{2} + p^{14} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 820610782 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 45740 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 4180097330 T^{2} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 38646 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 192224 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 27088624150 T^{2} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 86010 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 527419709110 T^{2} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 651718222942 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 301697627402 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 198996 T + p^{7} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 1209782 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 11632279636102 T^{2} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4939320 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 20468320226638 T^{2} + p^{14} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 6559712 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 44610274690150 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 5542410 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 141226276840510 T^{2} + p^{14} T^{4} \) |
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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
−10.76906494901858771245694910310, −10.11463868343549283381309888203, −10.09459797260717255010649693051, −9.049374922894600114657779551395, −9.005649381760983986315116627127, −8.279847348406947424589690991396, −8.250750051892631618612588855806, −7.21086518412152534763212197623, −6.87365505032953280793097407868, −6.21694617355284905994177329483, −6.12563332546728551381060931684, −5.37585232390751436182965304478, −4.44280018717106767057893116779, −4.20898812823277114484609747080, −3.86461166914105312522567239971, −2.65372189704082889585627911607, −2.60450357789736043370749309607, −1.55041541305374064807986350567, −1.20495237535909148603114176521, −0.16305338271400120926700042224,
0.16305338271400120926700042224, 1.20495237535909148603114176521, 1.55041541305374064807986350567, 2.60450357789736043370749309607, 2.65372189704082889585627911607, 3.86461166914105312522567239971, 4.20898812823277114484609747080, 4.44280018717106767057893116779, 5.37585232390751436182965304478, 6.12563332546728551381060931684, 6.21694617355284905994177329483, 6.87365505032953280793097407868, 7.21086518412152534763212197623, 8.250750051892631618612588855806, 8.279847348406947424589690991396, 9.005649381760983986315116627127, 9.049374922894600114657779551395, 10.09459797260717255010649693051, 10.11463868343549283381309888203, 10.76906494901858771245694910310
