| L(s) = 1 | − 4-s − 3·9-s − 8·11-s + 16-s − 2·19-s + 6·29-s − 4·31-s + 3·36-s − 12·41-s + 8·44-s − 11·49-s + 18·59-s − 24·61-s − 64-s + 2·76-s + 4·79-s − 4·89-s + 24·99-s − 16·101-s − 38·109-s − 6·116-s + 26·121-s + 4·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 9-s − 2.41·11-s + 1/4·16-s − 0.458·19-s + 1.11·29-s − 0.718·31-s + 1/2·36-s − 1.87·41-s + 1.20·44-s − 1.57·49-s + 2.34·59-s − 3.07·61-s − 1/8·64-s + 0.229·76-s + 0.450·79-s − 0.423·89-s + 2.41·99-s − 1.59·101-s − 3.63·109-s − 0.557·116-s + 2.36·121-s + 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3270177496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3270177496\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 63 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
| 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
−10.15016545704303514162933428331, −10.13005754909113889890701048623, −9.244549850815312309721075645447, −9.093704631429224472444185127512, −8.340239925267233839920725068411, −8.267657670479892378188304515591, −7.900926595129397868586337442323, −7.48816219465705187636854610825, −6.64126552377188817121822222429, −6.59378533420009187135711703750, −5.72012460110250655705893751189, −5.33685527201188894107005962850, −5.21063747931156207296261021631, −4.59185789638246659387353467167, −4.06614574231571688955308805184, −3.17293555578459101267045758585, −2.98335211476737464017735058856, −2.39703959808082716387771161589, −1.60717875927706914596224029356, −0.25849936757337310997818913150,
0.25849936757337310997818913150, 1.60717875927706914596224029356, 2.39703959808082716387771161589, 2.98335211476737464017735058856, 3.17293555578459101267045758585, 4.06614574231571688955308805184, 4.59185789638246659387353467167, 5.21063747931156207296261021631, 5.33685527201188894107005962850, 5.72012460110250655705893751189, 6.59378533420009187135711703750, 6.64126552377188817121822222429, 7.48816219465705187636854610825, 7.900926595129397868586337442323, 8.267657670479892378188304515591, 8.340239925267233839920725068411, 9.093704631429224472444185127512, 9.244549850815312309721075645447, 10.13005754909113889890701048623, 10.15016545704303514162933428331
