| L(s) = 1 | + 14·5-s − 22·7-s − 14·9-s − 6·11-s − 34·17-s + 38·19-s − 4·23-s + 97·25-s − 308·35-s + 42·43-s − 196·45-s + 10·47-s + 265·49-s − 84·55-s + 46·61-s + 308·63-s + 78·73-s + 132·77-s + 115·81-s + 12·83-s − 476·85-s + 532·95-s + 84·99-s + 244·101-s − 56·115-s + 748·119-s − 215·121-s + ⋯ |
| L(s) = 1 | + 14/5·5-s − 3.14·7-s − 1.55·9-s − 0.545·11-s − 2·17-s + 2·19-s − 0.173·23-s + 3.87·25-s − 8.79·35-s + 0.976·43-s − 4.35·45-s + 0.212·47-s + 5.40·49-s − 1.52·55-s + 0.754·61-s + 44/9·63-s + 1.06·73-s + 12/7·77-s + 1.41·81-s + 0.144·83-s − 5.59·85-s + 28/5·95-s + 0.848·99-s + 2.41·101-s − 0.486·115-s + 44/7·119-s − 1.77·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.443648992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.443648992\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 2 T + p^{2} T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 11 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 210 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 114 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 1890 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 1170 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 1794 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 21 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 5586 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5810 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 23 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 7410 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 1890 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 39 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 3234 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1730 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 94 T + p^{2} T^{2} )( 1 + 94 T + p^{2} T^{2} ) \) |
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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
−11.72716990787394579946016576959, −11.17748882824230981855980464195, −10.54677783313115130288056829326, −10.06327399832779005436961318447, −9.920648190491723612111621796326, −9.251641659973983475716296539917, −9.183373845964901131794394278504, −8.960601531284271578892874143964, −7.947479887722811995595826322818, −6.92909090305465226262312338300, −6.74564257609274195376491783404, −6.16493352664085345441282436373, −5.84228597035614170736785318061, −5.62765539636437164865864706312, −4.94547354567421502570066171107, −3.66174861071330556093889160267, −3.00073195545407294940842122854, −2.62365240655935044111853774613, −2.14103534836559364074491565415, −0.54072242621020803135999853043,
0.54072242621020803135999853043, 2.14103534836559364074491565415, 2.62365240655935044111853774613, 3.00073195545407294940842122854, 3.66174861071330556093889160267, 4.94547354567421502570066171107, 5.62765539636437164865864706312, 5.84228597035614170736785318061, 6.16493352664085345441282436373, 6.74564257609274195376491783404, 6.92909090305465226262312338300, 7.947479887722811995595826322818, 8.960601531284271578892874143964, 9.183373845964901131794394278504, 9.251641659973983475716296539917, 9.920648190491723612111621796326, 10.06327399832779005436961318447, 10.54677783313115130288056829326, 11.17748882824230981855980464195, 11.72716990787394579946016576959
