| L(s) = 1 | + 8·2-s + 18·3-s + 48·4-s + 50·5-s + 144·6-s − 98·7-s + 256·8-s + 243·9-s + 400·10-s + 198·11-s + 864·12-s + 866·13-s − 784·14-s + 900·15-s + 1.28e3·16-s + 404·17-s + 1.94e3·18-s + 2.09e3·19-s + 2.40e3·20-s − 1.76e3·21-s + 1.58e3·22-s + 5.07e3·23-s + 4.60e3·24-s + 1.87e3·25-s + 6.92e3·26-s + 2.91e3·27-s − 4.70e3·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s − 0.755·7-s + 1.41·8-s + 9-s + 1.26·10-s + 0.493·11-s + 1.73·12-s + 1.42·13-s − 1.06·14-s + 1.03·15-s + 5/4·16-s + 0.339·17-s + 1.41·18-s + 1.33·19-s + 1.34·20-s − 0.872·21-s + 0.697·22-s + 2.00·23-s + 1.63·24-s + 3/5·25-s + 2.00·26-s + 0.769·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(19.65194014\) |
| \(L(\frac12)\) |
\(\approx\) |
\(19.65194014\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 - 18 p T + 24474 p T^{2} - 18 p^{6} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 866 T + 365874 T^{2} - 866 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 404 T - 1131578 T^{2} - 404 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2098 T + 4485374 T^{2} - 2098 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 5076 T + 13045230 T^{2} - 5076 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5964 T + 45902526 T^{2} - 5964 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3422 T + 27023342 T^{2} + 3422 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2680 T + 49960598 T^{2} + 2680 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2696 T + 233278750 T^{2} - 2696 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5840 T + 293516070 T^{2} + 5840 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 9328 T + 476430814 T^{2} + 9328 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 14494 T + 793560026 T^{2} - 14494 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 9524 T + 1446256342 T^{2} - 9524 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 34604 T + 1731777662 T^{2} - 34604 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 33936 T + 957039638 T^{2} + 33936 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 10814 T + 2837218510 T^{2} + 10814 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3934 T + 3510521786 T^{2} - 3934 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 32656 T + 79446594 p T^{2} + 32656 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1360 T + 7877540662 T^{2} - 1360 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 41740 T + 3479181398 T^{2} + 41740 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 40378 T + 16633854354 T^{2} - 40378 p^{5} T^{3} + p^{10} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.88394862040155983906893918048, −11.38239303118127765212914637978, −10.76406096676586156347583352152, −10.34607938288086914383771354351, −9.689908194045967988578939652230, −9.395827559956657172711752482586, −8.625671995773432476381140819771, −8.484460125446985299480294584562, −7.27794343159124945734118431056, −7.23877391375484343775801212917, −6.28621554318474092138160114903, −6.28304208001201271735564923668, −5.13684590209275510963268068442, −5.11580907336123547137996134933, −3.77415034164177906093299792431, −3.74994095534974859703196235290, −2.75967811520726310919167721536, −2.74154259691033163839374423380, −1.27716331020885097066701593670, −1.26998026438961435546743732349,
1.26998026438961435546743732349, 1.27716331020885097066701593670, 2.74154259691033163839374423380, 2.75967811520726310919167721536, 3.74994095534974859703196235290, 3.77415034164177906093299792431, 5.11580907336123547137996134933, 5.13684590209275510963268068442, 6.28304208001201271735564923668, 6.28621554318474092138160114903, 7.23877391375484343775801212917, 7.27794343159124945734118431056, 8.484460125446985299480294584562, 8.625671995773432476381140819771, 9.395827559956657172711752482586, 9.689908194045967988578939652230, 10.34607938288086914383771354351, 10.76406096676586156347583352152, 11.38239303118127765212914637978, 11.88394862040155983906893918048
