L(s) = 1 | − 10·5-s + 2·7-s − 42·11-s − 14·13-s + 24·17-s + 140·19-s − 126·23-s + 75·25-s − 126·29-s + 56·31-s − 20·35-s − 572·37-s − 66·41-s + 530·43-s + 396·47-s − 494·49-s − 504·53-s + 420·55-s − 438·59-s − 602·61-s + 140·65-s + 920·67-s − 798·71-s − 770·73-s − 84·77-s + 1.72e3·79-s + 486·83-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 0.107·7-s − 1.15·11-s − 0.298·13-s + 0.342·17-s + 1.69·19-s − 1.14·23-s + 3/5·25-s − 0.806·29-s + 0.324·31-s − 0.0965·35-s − 2.54·37-s − 0.251·41-s + 1.87·43-s + 1.22·47-s − 1.44·49-s − 1.30·53-s + 1.02·55-s − 0.966·59-s − 1.26·61-s + 0.267·65-s + 1.67·67-s − 1.33·71-s − 1.23·73-s − 0.124·77-s + 2.45·79-s + 0.642·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4665600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4665600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8166027828\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8166027828\) |
\(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
good | 7 | $D_{4}$ | \( 1 - 2 T + 498 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 42 T + 2914 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 14 T - 282 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 24 T + 709 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 140 T + 18429 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 126 T + 16207 T^{2} + 126 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 126 T + 29878 T^{2} + 126 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 56 T + 5745 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 286 T + p^{3} T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 66 T + 55582 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 530 T + 219978 T^{2} - 530 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 396 T + 209806 T^{2} - 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 504 T + 243133 T^{2} + 504 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 438 T + 449458 T^{2} + 438 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 602 T + 211167 T^{2} + 602 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 920 T + 479730 T^{2} - 920 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 798 T + 832498 T^{2} + 798 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 770 T + 788478 T^{2} + 770 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1724 T + 1645773 T^{2} - 1724 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 486 T + 1054447 T^{2} - 486 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 294 T + 681406 T^{2} + 294 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 112 T + 618882 T^{2} - 112 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.749016548482755205982913833402, −8.628442886736332769039368087324, −7.83580942002962935874840310454, −7.77489384298858247854903252706, −7.48002417380488389910445260408, −7.29576781877854795537829382895, −6.48166615628107344836457939849, −6.27366145499345130168390591261, −5.53745967154544629117673218131, −5.43300764934482087129222436746, −4.77455970135566136532238914649, −4.73258208864253161497545598446, −3.77126314787628267190291023131, −3.73650548552533821513075148450, −3.03143125922218684439054559202, −2.82452064750427295288237937925, −2.00073435894270735420874362159, −1.60705514738449979110797926785, −0.815837609785272161945013762832, −0.22924545045305409245544262260,
0.22924545045305409245544262260, 0.815837609785272161945013762832, 1.60705514738449979110797926785, 2.00073435894270735420874362159, 2.82452064750427295288237937925, 3.03143125922218684439054559202, 3.73650548552533821513075148450, 3.77126314787628267190291023131, 4.73258208864253161497545598446, 4.77455970135566136532238914649, 5.43300764934482087129222436746, 5.53745967154544629117673218131, 6.27366145499345130168390591261, 6.48166615628107344836457939849, 7.29576781877854795537829382895, 7.48002417380488389910445260408, 7.77489384298858247854903252706, 7.83580942002962935874840310454, 8.628442886736332769039368087324, 8.749016548482755205982913833402