| L(s) = 1 | + 10·5-s + 14·7-s + 22·11-s − 24·13-s + 150·17-s + 8·19-s + 82·23-s − 158·25-s + 36·29-s + 88·31-s + 140·35-s + 288·37-s + 386·41-s − 344·43-s + 276·47-s + 147·49-s − 160·53-s + 220·55-s + 1.07e3·59-s + 156·61-s − 240·65-s − 1.37e3·67-s + 1.10e3·71-s − 240·73-s + 308·77-s − 412·79-s + 464·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s + 0.603·11-s − 0.512·13-s + 2.14·17-s + 0.0965·19-s + 0.743·23-s − 1.26·25-s + 0.230·29-s + 0.509·31-s + 0.676·35-s + 1.27·37-s + 1.47·41-s − 1.21·43-s + 0.856·47-s + 3/7·49-s − 0.414·53-s + 0.539·55-s + 2.37·59-s + 0.327·61-s − 0.457·65-s − 2.50·67-s + 1.84·71-s − 0.384·73-s + 0.455·77-s − 0.586·79-s + 0.613·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.444258427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.444258427\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 p T + 258 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 p T + 6 p^{2} T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 24 T + 4470 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 150 T + 13394 T^{2} - 150 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T - 3674 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 82 T + 7502 T^{2} - 82 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 36 T + 31694 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 88 T + 54718 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 288 T + 118710 T^{2} - 288 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 386 T + 111834 T^{2} - 386 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 p T + 90406 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 276 T + 20990 T^{2} - 276 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 160 T + 298646 T^{2} + 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1076 T + 664230 T^{2} - 1076 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 156 T + 87678 T^{2} - 156 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1372 T + 1071510 T^{2} + 1372 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1102 T + 967998 T^{2} - 1102 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 240 T + 253806 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 412 T + 837502 T^{2} + 412 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 464 T + 740166 T^{2} - 464 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1746 T + 2127850 T^{2} - 1746 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 856 T + 1966030 T^{2} - 856 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.830183938369148840846609257645, −8.828671976621766494521190268471, −8.055120170022899200702391158929, −7.80813358784054880987213107259, −7.54327326752670182360518261318, −7.13192432053686677364676008980, −6.56946296954006876338033236536, −6.06968434857259673197084233453, −5.81854995707628565092156273029, −5.51755153721745708290138704760, −4.82693729449203316149012437422, −4.81900757321163248850418110166, −3.91974214080510040959405071394, −3.77321487032999990169093780105, −3.01618248803347044867030311315, −2.62946950615073490791169022553, −1.99765387983941803737999822760, −1.62394635068244820258806150871, −0.887408232384105253740566607742, −0.70460359756030384268450851110,
0.70460359756030384268450851110, 0.887408232384105253740566607742, 1.62394635068244820258806150871, 1.99765387983941803737999822760, 2.62946950615073490791169022553, 3.01618248803347044867030311315, 3.77321487032999990169093780105, 3.91974214080510040959405071394, 4.81900757321163248850418110166, 4.82693729449203316149012437422, 5.51755153721745708290138704760, 5.81854995707628565092156273029, 6.06968434857259673197084233453, 6.56946296954006876338033236536, 7.13192432053686677364676008980, 7.54327326752670182360518261318, 7.80813358784054880987213107259, 8.055120170022899200702391158929, 8.828671976621766494521190268471, 8.830183938369148840846609257645
