| L(s) = 1 | − 729·9-s + 6.91e3·11-s + 2.60e3·19-s − 3.48e5·29-s + 3.79e5·31-s + 7.15e5·41-s + 1.63e6·49-s + 1.07e6·59-s − 3.89e6·61-s + 3.36e6·71-s + 9.13e6·79-s + 5.31e5·81-s + 1.09e7·89-s − 5.03e6·99-s + 9.74e6·101-s + 2.92e7·109-s − 3.14e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.04e8·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 1.56·11-s + 0.0869·19-s − 2.65·29-s + 2.28·31-s + 1.62·41-s + 1.98·49-s + 0.684·59-s − 2.19·61-s + 1.11·71-s + 2.08·79-s + 1/9·81-s + 1.64·89-s − 0.521·99-s + 0.940·101-s + 2.16·109-s − 0.161·121-s + 1.66·169-s − 0.0289·171-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
| \(L(4)\) |
\(\approx\) |
\(4.691488797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.691488797\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2^2$ | \( 1 - 1638622 T^{2} + p^{14} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3456 T + p^{7} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 104244934 T^{2} + p^{14} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 514357342 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 1300 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6611179150 T^{2} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 174306 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 189824 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 111740150230 T^{2} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 357690 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 463065264310 T^{2} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 1002905791582 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2195307934198 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 539904 T + p^{7} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 1946338 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 8677236564742 T^{2} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 1683840 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 976227982 p^{2} T^{2} + p^{14} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4565008 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 24625514441830 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 5461230 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 15798526957630 T^{2} + p^{14} T^{4} \) |
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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
−10.60282362569688836298069105519, −10.52057873567206715162457179732, −9.484380892676338418022296236441, −9.440645416229341230870377766692, −9.016962196573957398644722221476, −8.462097296721489447506836366642, −7.70148794794312966462592673956, −7.60479304189960627343238506019, −6.69341478070357782469302133665, −6.50628567296527078183623590571, −5.71640381727000887709829316435, −5.57286522372213631971866206824, −4.44977118452718504876886143657, −4.33510406645739152403300837565, −3.55076873289683500970335193313, −3.11963947616300108787515494721, −2.18367721922945126186860438806, −1.82668293063021851022547760653, −0.73464640242532934006281022375, −0.72415123455300507456336497161,
0.72415123455300507456336497161, 0.73464640242532934006281022375, 1.82668293063021851022547760653, 2.18367721922945126186860438806, 3.11963947616300108787515494721, 3.55076873289683500970335193313, 4.33510406645739152403300837565, 4.44977118452718504876886143657, 5.57286522372213631971866206824, 5.71640381727000887709829316435, 6.50628567296527078183623590571, 6.69341478070357782469302133665, 7.60479304189960627343238506019, 7.70148794794312966462592673956, 8.462097296721489447506836366642, 9.016962196573957398644722221476, 9.440645416229341230870377766692, 9.484380892676338418022296236441, 10.52057873567206715162457179732, 10.60282362569688836298069105519
