| L(s) = 1 | + 2-s + 3·3-s − 6·5-s + 3·6-s − 8-s + 6·9-s − 6·10-s − 6·11-s − 13-s − 18·15-s − 16-s + 3·17-s + 6·18-s − 7·19-s − 6·22-s − 18·23-s − 3·24-s + 17·25-s − 26-s + 9·27-s − 3·29-s − 18·30-s + 8·31-s − 18·33-s + 3·34-s + 37-s − 7·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s − 2.68·5-s + 1.22·6-s − 0.353·8-s + 2·9-s − 1.89·10-s − 1.80·11-s − 0.277·13-s − 4.64·15-s − 1/4·16-s + 0.727·17-s + 1.41·18-s − 1.60·19-s − 1.27·22-s − 3.75·23-s − 0.612·24-s + 17/5·25-s − 0.196·26-s + 1.73·27-s − 0.557·29-s − 3.28·30-s + 1.43·31-s − 3.13·33-s + 0.514·34-s + 0.164·37-s − 1.13·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.084121015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.084121015\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−10.40566914714433396709764231678, −9.768264604629468990740190544816, −9.708241134103259692873776403241, −8.735121365276763786226819668956, −8.433707218054355096952850014232, −8.008555520496974770775095554190, −7.957909151609678218279155514398, −7.67725208445889410329675110591, −7.35905522455404569631532918585, −6.34612927954513809526707694918, −6.26627247333053901676351892004, −5.19078692929344181832009996439, −4.86299925282796577590166867486, −4.05717899331588196708266165056, −3.99550013219839204136959435936, −3.77037182893131441837864322630, −3.11074172310220503733205894743, −2.35647720321303614509890025667, −2.18821681019475925448855148512, −0.37947664452942209646127935149,
0.37947664452942209646127935149, 2.18821681019475925448855148512, 2.35647720321303614509890025667, 3.11074172310220503733205894743, 3.77037182893131441837864322630, 3.99550013219839204136959435936, 4.05717899331588196708266165056, 4.86299925282796577590166867486, 5.19078692929344181832009996439, 6.26627247333053901676351892004, 6.34612927954513809526707694918, 7.35905522455404569631532918585, 7.67725208445889410329675110591, 7.957909151609678218279155514398, 8.008555520496974770775095554190, 8.433707218054355096952850014232, 8.735121365276763786226819668956, 9.708241134103259692873776403241, 9.768264604629468990740190544816, 10.40566914714433396709764231678
