| 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
| \(L(2)\) |
\(\approx\) |
\(3.207978613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.207978613\) |
| \(L(\frac{5}{2})\) |
|
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
−9.118144728163305515599355323924, −8.667998735596159592096954354947, −7.997805238252674830794961245943, −7.82751637766318520742778509604, −7.39812811324360098142376256970, −7.32201964606253524831679544693, −6.29818585102198258603901547334, −6.20569148686128275953609530846, −5.80539062907689577760967425890, −5.66723131215722492867239808608, −4.93100765000859982280196631434, −4.69186790253580034533868320609, −3.91698269215146795822437845928, −3.65591303610411636571459678125, −2.98826125209207776141205490801, −2.72401274728565102551113068820, −2.04777074903997506085557718340, −1.71540018145978445777631758291, −0.851329705570831133563955842902, −0.44454467361667740233757987322,
0.44454467361667740233757987322, 0.851329705570831133563955842902, 1.71540018145978445777631758291, 2.04777074903997506085557718340, 2.72401274728565102551113068820, 2.98826125209207776141205490801, 3.65591303610411636571459678125, 3.91698269215146795822437845928, 4.69186790253580034533868320609, 4.93100765000859982280196631434, 5.66723131215722492867239808608, 5.80539062907689577760967425890, 6.20569148686128275953609530846, 6.29818585102198258603901547334, 7.32201964606253524831679544693, 7.39812811324360098142376256970, 7.82751637766318520742778509604, 7.997805238252674830794961245943, 8.667998735596159592096954354947, 9.118144728163305515599355323924
