| L(s) = 1 | + 2·2-s + 12·3-s + 2·4-s + 24·6-s + 52·7-s + 32·8-s + 72·9-s − 16·11-s + 24·12-s − 278·13-s + 104·14-s − 132·16-s + 2·17-s + 144·18-s + 624·21-s − 32·22-s + 332·23-s + 384·24-s − 556·26-s + 972·27-s + 104·28-s + 1.14e3·31-s − 776·32-s − 192·33-s + 4·34-s + 144·36-s + 502·37-s + ⋯ |
| L(s) = 1 | + 1/2·2-s + 4/3·3-s + 1/8·4-s + 2/3·6-s + 1.06·7-s + 1/2·8-s + 8/9·9-s − 0.132·11-s + 1/6·12-s − 1.64·13-s + 0.530·14-s − 0.515·16-s + 0.00692·17-s + 4/9·18-s + 1.41·21-s − 0.0661·22-s + 0.627·23-s + 2/3·24-s − 0.822·26-s + 4/3·27-s + 0.132·28-s + 1.19·31-s − 0.757·32-s − 0.176·33-s + 0.00346·34-s + 1/9·36-s + 0.366·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.192205745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.192205745\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{5} T^{3} + p^{8} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 4 p T + 8 p^{2} T^{2} - 4 p^{5} T^{3} + p^{8} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 52 T + 1352 T^{2} - 52 p^{4} T^{3} + p^{8} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 8 T + p^{4} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 278 T + 38642 T^{2} + 278 p^{4} T^{3} + p^{8} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p^{4} T^{3} + p^{8} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 228242 T^{2} + p^{8} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 332 T + 55112 T^{2} - 332 p^{4} T^{3} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1184162 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 572 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 502 T + 126002 T^{2} - 502 p^{4} T^{3} + p^{8} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 1688 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 2948 T + 4345352 T^{2} + 2948 p^{4} T^{3} + p^{8} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4948 T + 12241352 T^{2} + 4948 p^{4} T^{3} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6662 T + 22191122 T^{2} - 6662 p^{4} T^{3} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10839122 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 1592 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 1748 T + 1527752 T^{2} + 1748 p^{4} T^{3} + p^{8} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6068 T + p^{4} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 1582 T + 1251362 T^{2} - 1582 p^{4} T^{3} + p^{8} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 5274238 T^{2} + p^{8} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 11308 T + 63935432 T^{2} + 11308 p^{4} T^{3} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 120818882 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 13102 T + 85831202 T^{2} - 13102 p^{4} T^{3} + p^{8} T^{4} \) |
| 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
−16.96753199929493457223159140015, −16.57645246817974312402098165978, −15.54613954379934164481795409562, −15.05463991506317715010787545683, −14.42678488507848745874588254667, −14.30728284393453595601645287190, −13.24154376785233251414185884472, −13.18317469276499831588352636635, −11.83242685925812946990721583049, −11.64473758892957704641535184478, −10.33037718026605705776005055519, −9.947963772307172587935621680019, −8.763389976719855044087727291287, −8.360832363864747618605310015903, −7.45483687878379905933832580018, −6.81729252802811190349933210556, −5.01165484082599337288785049667, −4.60825234055435024188664084364, −3.07928519069609438437336194380, −1.97588809027840070955203884040,
1.97588809027840070955203884040, 3.07928519069609438437336194380, 4.60825234055435024188664084364, 5.01165484082599337288785049667, 6.81729252802811190349933210556, 7.45483687878379905933832580018, 8.360832363864747618605310015903, 8.763389976719855044087727291287, 9.947963772307172587935621680019, 10.33037718026605705776005055519, 11.64473758892957704641535184478, 11.83242685925812946990721583049, 13.18317469276499831588352636635, 13.24154376785233251414185884472, 14.30728284393453595601645287190, 14.42678488507848745874588254667, 15.05463991506317715010787545683, 15.54613954379934164481795409562, 16.57645246817974312402098165978, 16.96753199929493457223159140015
