| L(s) = 1 | − 3-s − 7-s − 9-s + 3·11-s − 7·19-s + 21-s − 6·23-s − 4·25-s + 10·29-s + 6·31-s − 3·33-s − 8·37-s + 2·41-s − 3·43-s − 4·47-s − 4·49-s + 4·53-s + 7·57-s − 5·59-s − 16·61-s + 63-s + 11·67-s + 6·69-s − 8·71-s + 6·73-s + 4·75-s − 3·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 1/3·9-s + 0.904·11-s − 1.60·19-s + 0.218·21-s − 1.25·23-s − 4/5·25-s + 1.85·29-s + 1.07·31-s − 0.522·33-s − 1.31·37-s + 0.312·41-s − 0.457·43-s − 0.583·47-s − 4/7·49-s + 0.549·53-s + 0.927·57-s − 0.650·59-s − 2.04·61-s + 0.125·63-s + 1.34·67-s + 0.722·69-s − 0.949·71-s + 0.702·73-s + 0.461·75-s − 0.341·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100240 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100240 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 179 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 24 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 16 T + 158 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 11 T + 94 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 + 7 T + 106 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 11 T + 102 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - T + 84 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 19 T + 210 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
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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
−14.3181699164, −13.7610296326, −13.5539198723, −12.8510534438, −12.4029330909, −11.9830469374, −11.8211718914, −11.2171832038, −10.7503461474, −10.1489218093, −10.0182523289, −9.37806877483, −8.65367270920, −8.50271718957, −7.92327209991, −7.24329403177, −6.45816827684, −6.36796877995, −5.95753288476, −5.13359374459, −4.49187207201, −4.04586992578, −3.29032004030, −2.45308235531, −1.53470252639, 0,
1.53470252639, 2.45308235531, 3.29032004030, 4.04586992578, 4.49187207201, 5.13359374459, 5.95753288476, 6.36796877995, 6.45816827684, 7.24329403177, 7.92327209991, 8.50271718957, 8.65367270920, 9.37806877483, 10.0182523289, 10.1489218093, 10.7503461474, 11.2171832038, 11.8211718914, 11.9830469374, 12.4029330909, 12.8510534438, 13.5539198723, 13.7610296326, 14.3181699164
