| L(s) = 1 | − 16·4-s + 477·9-s + 810·11-s + 256·16-s − 4.68e3·19-s − 1.65e4·29-s + 8.03e3·31-s − 7.63e3·36-s + 6.67e3·41-s − 1.29e4·44-s − 2.40e3·49-s + 3.76e4·59-s + 4.33e4·61-s − 4.09e3·64-s − 5.71e4·71-s + 7.49e4·76-s − 1.17e5·79-s + 1.68e5·81-s − 2.70e5·89-s + 3.86e5·99-s + 6.69e4·101-s + 3.02e5·109-s + 2.64e5·116-s + 1.69e5·121-s − 1.28e5·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.96·9-s + 2.01·11-s + 1/4·16-s − 2.97·19-s − 3.64·29-s + 1.50·31-s − 0.981·36-s + 0.619·41-s − 1.00·44-s − 1/7·49-s + 1.40·59-s + 1.49·61-s − 1/8·64-s − 1.34·71-s + 1.48·76-s − 2.12·79-s + 2.85·81-s − 3.62·89-s + 3.96·99-s + 0.652·101-s + 2.43·109-s + 1.82·116-s + 1.05·121-s − 0.750·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.407016997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.407016997\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 53 p^{2} T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 405 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 589705 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 1841713 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2342 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6967786 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8259 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4016 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 89098150 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3336 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 259079438 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 352249525 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 826879930 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 18816 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 21668 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 16661162 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 28560 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 801853778 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 58823 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 7877509750 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 135384 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 4978791289 T^{2} + p^{10} T^{4} \) |
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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
−10.87524964580864360048042156699, −10.30746709885766710239771793117, −9.924204737832344596010864951707, −9.500413462088595009244174372353, −9.147879500394021129993785859046, −8.530502277944782146789754948484, −8.307446878714609250873939286612, −7.41624060770527472970787346996, −6.92158222362119014219056018762, −6.84495175321044538955280963000, −5.97913053468422386248293348860, −5.75913212469939274370912961172, −4.52223876740094904572226963669, −4.41217257539128596325672669537, −3.93556482073802739511391746779, −3.61827897192229791230810929295, −2.28889899083210710493360341396, −1.74760237765255382078927926585, −1.29871180177463202266503025227, −0.41671397945096206055179035081,
0.41671397945096206055179035081, 1.29871180177463202266503025227, 1.74760237765255382078927926585, 2.28889899083210710493360341396, 3.61827897192229791230810929295, 3.93556482073802739511391746779, 4.41217257539128596325672669537, 4.52223876740094904572226963669, 5.75913212469939274370912961172, 5.97913053468422386248293348860, 6.84495175321044538955280963000, 6.92158222362119014219056018762, 7.41624060770527472970787346996, 8.307446878714609250873939286612, 8.530502277944782146789754948484, 9.147879500394021129993785859046, 9.500413462088595009244174372353, 9.924204737832344596010864951707, 10.30746709885766710239771793117, 10.87524964580864360048042156699
