L(s) = 1 | + 162·9-s + 1.31e3·11-s + 2.72e3·19-s + 4.43e3·29-s − 3.40e3·31-s − 3.63e3·41-s − 2.49e4·49-s − 1.73e4·59-s − 6.93e4·61-s + 1.89e3·71-s − 9.30e4·79-s − 3.28e4·81-s + 2.09e5·89-s + 2.12e5·99-s + 8.49e4·101-s + 2.90e5·109-s + 9.68e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.00e5·169-s + ⋯ |
L(s) = 1 | + 2/3·9-s + 3.26·11-s + 1.73·19-s + 0.979·29-s − 0.635·31-s − 0.337·41-s − 1.48·49-s − 0.648·59-s − 2.38·61-s + 0.0446·71-s − 1.67·79-s − 5/9·81-s + 2.80·89-s + 2.17·99-s + 0.828·101-s + 2.33·109-s + 6.01·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 1.88·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(4.617318054\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.617318054\) |
\(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 - 2 p^{4} T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 24950 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 656 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 700150 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16386 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 1364 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 8041482 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2218 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 1700 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 137972198 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 1818 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 183051730 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 312908538 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 225456410 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8668 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 34670 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 437725858 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 948 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 164280782 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 46536 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 3451998 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 104934 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 15860327998 T^{2} + p^{10} T^{4} \) |
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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
−11.96055177592254939183594249237, −11.41557916868221746021688632943, −11.03925985308613112581005577329, −10.17049105441251841046832650518, −9.813239160732917343498780638489, −9.205397388826894397048941373039, −9.112415629263569486379147222845, −8.453092380269791844209022712527, −7.58603119174489860603208742333, −7.24781720247463542924883917758, −6.57724988554881623945761834069, −6.31107529601396488870191056785, −5.63093695567149132903309164600, −4.56311255729882786575800860023, −4.45329224467765963188390326017, −3.36030051429201939748214060044, −3.35065002446976883109434462910, −1.76865738407158621027249172114, −1.38675957037790111408376744312, −0.74123649422388518967884185223,
0.74123649422388518967884185223, 1.38675957037790111408376744312, 1.76865738407158621027249172114, 3.35065002446976883109434462910, 3.36030051429201939748214060044, 4.45329224467765963188390326017, 4.56311255729882786575800860023, 5.63093695567149132903309164600, 6.31107529601396488870191056785, 6.57724988554881623945761834069, 7.24781720247463542924883917758, 7.58603119174489860603208742333, 8.453092380269791844209022712527, 9.112415629263569486379147222845, 9.205397388826894397048941373039, 9.813239160732917343498780638489, 10.17049105441251841046832650518, 11.03925985308613112581005577329, 11.41557916868221746021688632943, 11.96055177592254939183594249237