| L(s) = 1 | + 422·9-s + 1.20e3·11-s − 2.64e3·19-s − 1.18e4·29-s + 6.64e3·31-s − 3.59e4·41-s + 2.19e4·49-s + 6.64e4·59-s − 8.04e4·61-s + 1.10e5·71-s + 6.29e4·79-s + 1.19e5·81-s + 1.81e5·89-s + 5.09e5·99-s − 1.45e5·101-s + 1.40e5·109-s + 7.72e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6.48e5·169-s + ⋯ |
| L(s) = 1 | + 1.73·9-s + 3.01·11-s − 1.68·19-s − 2.60·29-s + 1.24·31-s − 3.33·41-s + 1.30·49-s + 2.48·59-s − 2.76·61-s + 2.60·71-s + 1.13·79-s + 2.01·81-s + 2.43·89-s + 5.22·99-s − 1.41·101-s + 1.13·109-s + 4.79·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 1.74·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.373197693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.373197693\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 422 T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 21950 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 604 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 648950 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 1974814 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 1324 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 12146782 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5902 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3320 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 22608838 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 438 p T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 208195190 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 362728398 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 151705370 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 33228 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 40210 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 764720282 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 55312 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3403144622 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 31456 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 7275280582 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 90854 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 6759265922 T^{2} + p^{10} T^{4} \) |
| 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
−10.65445035125637801813720762235, −10.21607833990852704940050712860, −9.546362931279703708393155210001, −9.534214231269797203751912698221, −8.835494624089443193764362778523, −8.613023420498874296604025759046, −7.86668449649405892437048487962, −7.30302542054420591552798372388, −6.71804863205368340943235155279, −6.61069484597969621770665758270, −6.23505761028207078459364958855, −5.33828625764272842774081439359, −4.68960456417940979107240878305, −4.16597265225463421745655187640, −3.70012032248649133552305494802, −3.60387233109569261711995451988, −2.10416239428805585811590575855, −1.78791649668315935550534200206, −1.26660297723123933781970023821, −0.54332976580278596911369038418,
0.54332976580278596911369038418, 1.26660297723123933781970023821, 1.78791649668315935550534200206, 2.10416239428805585811590575855, 3.60387233109569261711995451988, 3.70012032248649133552305494802, 4.16597265225463421745655187640, 4.68960456417940979107240878305, 5.33828625764272842774081439359, 6.23505761028207078459364958855, 6.61069484597969621770665758270, 6.71804863205368340943235155279, 7.30302542054420591552798372388, 7.86668449649405892437048487962, 8.613023420498874296604025759046, 8.835494624089443193764362778523, 9.534214231269797203751912698221, 9.546362931279703708393155210001, 10.21607833990852704940050712860, 10.65445035125637801813720762235
