| L(s) = 1 | + 8·3-s + 50·5-s + 104·7-s − 158·9-s + 320·11-s − 100·13-s + 400·15-s + 580·17-s + 720·19-s + 832·21-s + 1.68e3·23-s + 1.87e3·25-s − 1.09e3·27-s + 108·29-s + 9.84e3·31-s + 2.56e3·33-s + 5.20e3·35-s + 6.54e3·37-s − 800·39-s − 1.06e4·41-s + 2.56e4·43-s − 7.90e3·45-s + 2.82e4·47-s − 2.52e4·49-s + 4.64e3·51-s + 3.13e4·53-s + 1.60e4·55-s + ⋯ |
| L(s) = 1 | + 0.513·3-s + 0.894·5-s + 0.802·7-s − 0.650·9-s + 0.797·11-s − 0.164·13-s + 0.459·15-s + 0.486·17-s + 0.457·19-s + 0.411·21-s + 0.665·23-s + 3/5·25-s − 0.289·27-s + 0.0238·29-s + 1.83·31-s + 0.409·33-s + 0.717·35-s + 0.785·37-s − 0.0842·39-s − 0.986·41-s + 2.11·43-s − 0.581·45-s + 1.86·47-s − 1.50·49-s + 0.249·51-s + 1.53·53-s + 0.713·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.544146337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.544146337\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 8 T + 74 p T^{2} - 8 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 104 T + 36038 T^{2} - 104 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 320 T + 319702 T^{2} - 320 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 100 T + 297086 T^{2} + 100 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 580 T + 2475814 T^{2} - 580 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 720 T + 4633798 T^{2} - 720 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 1688 T + 10950502 T^{2} - 1688 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 108 T + 39233214 T^{2} - 108 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 9840 T + 76732702 T^{2} - 9840 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6540 T + 61572814 T^{2} - 6540 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10620 T + 77106582 T^{2} + 10620 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 25672 T + 364912302 T^{2} - 25672 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 28296 T + 617782998 T^{2} - 28296 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 31340 T + 755347886 T^{2} - 31340 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 30800 T + 1666896598 T^{2} - 30800 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 24540 T + 1447225822 T^{2} - 24540 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 34584 T + 2930656478 T^{2} - 34584 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12400 T + 1143670702 T^{2} + 12400 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7180 T + 284279286 T^{2} + 7180 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 71840 T + 7430807198 T^{2} - 71840 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 31928 T + 5910993662 T^{2} + 31928 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 40748 T + 8570866774 T^{2} + 40748 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 190140 T + 19653817414 T^{2} + 190140 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−12.10334818284488564526303324012, −11.82704543552943724380179264500, −11.18514327052685527468570444557, −10.81809875389543889540661304757, −10.01484352254298780920174565565, −9.769666204051376942254857200471, −9.049793603702300003964348872616, −8.762616218540841382037509222076, −8.121813769532523070459085255303, −7.70257791294607010306642768406, −6.83374691369088593480486999847, −6.47588285661736547919682466653, −5.49164078420755818348702884741, −5.40962893907297669090338215654, −4.42874427049231063031080235284, −3.81535126494829123129434427721, −2.70963162136766918405162495655, −2.50883897290513374612480264381, −1.34742039953208690732897125716, −0.847092076456144741237769181393,
0.847092076456144741237769181393, 1.34742039953208690732897125716, 2.50883897290513374612480264381, 2.70963162136766918405162495655, 3.81535126494829123129434427721, 4.42874427049231063031080235284, 5.40962893907297669090338215654, 5.49164078420755818348702884741, 6.47588285661736547919682466653, 6.83374691369088593480486999847, 7.70257791294607010306642768406, 8.121813769532523070459085255303, 8.762616218540841382037509222076, 9.049793603702300003964348872616, 9.769666204051376942254857200471, 10.01484352254298780920174565565, 10.81809875389543889540661304757, 11.18514327052685527468570444557, 11.82704543552943724380179264500, 12.10334818284488564526303324012
