| L(s) = 1 | − 4·4-s + 6·5-s − 18·9-s + 16·16-s − 24·20-s + 11·25-s + 84·29-s + 72·36-s − 36·41-s − 108·45-s − 98·49-s + 44·61-s − 64·64-s + 96·80-s + 243·81-s − 156·89-s − 44·100-s − 396·101-s + 364·109-s − 336·116-s + 242·121-s − 84·125-s + 127-s + 131-s + 137-s + 139-s − 288·144-s + ⋯ |
| L(s) = 1 | − 4-s + 6/5·5-s − 2·9-s + 16-s − 6/5·20-s + 0.439·25-s + 2.89·29-s + 2·36-s − 0.878·41-s − 2.39·45-s − 2·49-s + 0.721·61-s − 64-s + 6/5·80-s + 3·81-s − 1.75·89-s − 0.439·100-s − 3.92·101-s + 3.33·109-s − 2.89·116-s + 2·121-s − 0.671·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 2·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6871784578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6871784578\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 - 6 T + p^{2} T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )( 1 + 10 T + p^{2} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )( 1 + 30 T + p^{2} T^{2} ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )( 1 + 70 T + p^{2} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 90 T + p^{2} T^{2} )( 1 + 90 T + p^{2} T^{2} ) \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 110 T + p^{2} T^{2} )( 1 + 110 T + p^{2} T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 78 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 130 T + p^{2} T^{2} )( 1 + 130 T + p^{2} T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.96062590567325815104023523697, −17.94885027651970621775195863569, −17.33399256142956227684504519776, −16.87855723835633671836689974348, −16.12067171012368066821763565845, −15.02155578445027214922818563134, −14.34275859722948862111176454258, −13.91615602008035128526283404749, −13.59782577234706309083810554246, −12.62713949989021404985482988469, −11.90550244298984676775472499594, −11.02542479449272167448992573251, −10.09372891876765178042217088505, −9.547024869041151195264697437329, −8.520031008385668286043288259253, −8.324070771202873362775595250013, −6.50419717571286079875050761495, −5.70955971770895197519786291105, −4.87586661189760822228494762830, −2.98280952584571453573837281815,
2.98280952584571453573837281815, 4.87586661189760822228494762830, 5.70955971770895197519786291105, 6.50419717571286079875050761495, 8.324070771202873362775595250013, 8.520031008385668286043288259253, 9.547024869041151195264697437329, 10.09372891876765178042217088505, 11.02542479449272167448992573251, 11.90550244298984676775472499594, 12.62713949989021404985482988469, 13.59782577234706309083810554246, 13.91615602008035128526283404749, 14.34275859722948862111176454258, 15.02155578445027214922818563134, 16.12067171012368066821763565845, 16.87855723835633671836689974348, 17.33399256142956227684504519776, 17.94885027651970621775195863569, 17.96062590567325815104023523697
