L(s) = 1 | − 3-s + 3·5-s + 4·7-s + 3·9-s − 3·11-s − 3·15-s − 9·17-s − 7·19-s − 4·21-s + 15·23-s + 25-s − 8·27-s − 12·29-s − 5·31-s + 3·33-s + 12·35-s + 5·37-s + 9·45-s − 3·47-s + 9·49-s + 9·51-s + 9·53-s − 9·55-s + 7·57-s − 9·59-s + 15·61-s + 12·63-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.34·5-s + 1.51·7-s + 9-s − 0.904·11-s − 0.774·15-s − 2.18·17-s − 1.60·19-s − 0.872·21-s + 3.12·23-s + 1/5·25-s − 1.53·27-s − 2.22·29-s − 0.898·31-s + 0.522·33-s + 2.02·35-s + 0.821·37-s + 1.34·45-s − 0.437·47-s + 9/7·49-s + 1.26·51-s + 1.23·53-s − 1.21·55-s + 0.927·57-s − 1.17·59-s + 1.92·61-s + 1.51·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.167232804\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.167232804\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 9 T + 44 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 15 T + 98 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 9 T + 94 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 21 T + 236 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−13.48390054678584073455726518985, −13.36435495900951915893436555353, −12.90991557214120478155665202877, −12.74792982753816727685264134953, −11.35668188794049704482501219594, −11.29943998937957979654265389195, −10.77906089588489236258136790061, −10.53881676871788679563498254985, −9.391354172952845164126764313371, −9.343825111569200105677304811287, −8.572229715794336102069115036051, −7.894237687879887047672717842079, −6.99159975970254225762761686382, −6.85205033185553107187980022870, −5.74024338411001776721315514030, −5.35765362414082962117536760423, −4.71525060893812083005003990248, −4.07424526339612746354847297653, −2.29797226428550356109097376842, −1.82698880999477658675413412774,
1.82698880999477658675413412774, 2.29797226428550356109097376842, 4.07424526339612746354847297653, 4.71525060893812083005003990248, 5.35765362414082962117536760423, 5.74024338411001776721315514030, 6.85205033185553107187980022870, 6.99159975970254225762761686382, 7.894237687879887047672717842079, 8.572229715794336102069115036051, 9.343825111569200105677304811287, 9.391354172952845164126764313371, 10.53881676871788679563498254985, 10.77906089588489236258136790061, 11.29943998937957979654265389195, 11.35668188794049704482501219594, 12.74792982753816727685264134953, 12.90991557214120478155665202877, 13.36435495900951915893436555353, 13.48390054678584073455726518985