| L(s) = 1 | − 16·4-s − 384·11-s + 256·16-s + 5.48e3·19-s + 1.18e4·29-s − 1.37e4·31-s + 756·41-s + 6.14e3·44-s + 1.96e4·49-s − 6.99e4·59-s − 1.96e4·61-s − 4.09e3·64-s − 1.40e5·71-s − 8.76e4·76-s − 9.04e3·79-s + 7.69e4·89-s − 1.55e5·101-s − 4.13e5·109-s − 1.89e5·116-s − 2.11e5·121-s + 2.19e5·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.956·11-s + 1/4·16-s + 3.48·19-s + 2.60·29-s − 2.56·31-s + 0.0702·41-s + 0.478·44-s + 1.17·49-s − 2.61·59-s − 0.677·61-s − 1/8·64-s − 3.30·71-s − 1.74·76-s − 0.162·79-s + 1.03·89-s − 1.51·101-s − 3.33·109-s − 1.30·116-s − 1.31·121-s + 1.28·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.428823547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.428823547\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 19690 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 192 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 480650 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2259070 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2740 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10420330 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5910 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6868 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 108239590 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 378 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 288092530 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 286503130 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 752228710 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 34980 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 9838 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 1563076930 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 70212 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3662758990 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4520 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 4019056190 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 38490 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 17171001790 T^{2} + p^{10} T^{4} \) |
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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
−10.76656733887670664673149642774, −10.08241192306380424153451880554, −9.557455293700342535705976263121, −9.061787114554650116522998804702, −9.004276915383647133073234122526, −8.065596996378590770652163928973, −7.81100620797478577951178630402, −7.33386643894994979579433650161, −7.05922621110082968527470876357, −6.17541032039696109384632373287, −5.67673421656492059811411036886, −5.24077279646477479109744971849, −4.92878701657058044852398638875, −4.26548087935373380214980896378, −3.55652948421125300300334901107, −2.86157357112046419438571410788, −2.83459954782321005906137828440, −1.48842805208648658581262739744, −1.18624832241486541605750409412, −0.31281360253494608468916559672,
0.31281360253494608468916559672, 1.18624832241486541605750409412, 1.48842805208648658581262739744, 2.83459954782321005906137828440, 2.86157357112046419438571410788, 3.55652948421125300300334901107, 4.26548087935373380214980896378, 4.92878701657058044852398638875, 5.24077279646477479109744971849, 5.67673421656492059811411036886, 6.17541032039696109384632373287, 7.05922621110082968527470876357, 7.33386643894994979579433650161, 7.81100620797478577951178630402, 8.065596996378590770652163928973, 9.004276915383647133073234122526, 9.061787114554650116522998804702, 9.557455293700342535705976263121, 10.08241192306380424153451880554, 10.76656733887670664673149642774
