| L(s) = 1 | − 2·2-s − 12·3-s + 2·4-s + 40·5-s + 24·6-s − 52·7-s − 32·8-s + 72·9-s − 80·10-s − 16·11-s − 24·12-s + 278·13-s + 104·14-s − 480·15-s − 132·16-s − 2·17-s − 144·18-s + 80·20-s + 624·21-s + 32·22-s − 332·23-s + 384·24-s + 975·25-s − 556·26-s − 972·27-s − 104·28-s + 960·30-s + ⋯ |
| L(s) = 1 | − 1/2·2-s − 4/3·3-s + 1/8·4-s + 8/5·5-s + 2/3·6-s − 1.06·7-s − 1/2·8-s + 8/9·9-s − 4/5·10-s − 0.132·11-s − 1/6·12-s + 1.64·13-s + 0.530·14-s − 2.13·15-s − 0.515·16-s − 0.00692·17-s − 4/9·18-s + 1/5·20-s + 1.41·21-s + 0.0661·22-s − 0.627·23-s + 2/3·24-s + 1.55·25-s − 0.822·26-s − 4/3·27-s − 0.132·28-s + 1.06·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4484673455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4484673455\) |
| \(L(3)\) |
|
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 | $C_2$ | \( 1 - 8 p T + p^{4} T^{2} \) |
| 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
−23.89913624452159933516741890958, −23.26434389688325556266865988942, −22.28910821114774274241655878508, −22.15309634886302607912586360934, −20.88647769646167381974628978295, −20.63801047044014262264701318701, −18.91495198703454935137597208810, −18.50275335165158129876739462903, −17.41568124951225360560227193675, −17.40897162649728375522062915889, −16.16030001239991370174501813980, −15.64547499604769225484994205118, −13.78514932864529845373034703274, −13.23394830568435968840208799345, −12.03131972356520577702595263087, −10.84818845483526284000633729910, −9.959603234283732465813800828395, −8.992004666314940419028087030184, −6.43446901279239483854321391270, −5.87345398363083830302364819598,
5.87345398363083830302364819598, 6.43446901279239483854321391270, 8.992004666314940419028087030184, 9.959603234283732465813800828395, 10.84818845483526284000633729910, 12.03131972356520577702595263087, 13.23394830568435968840208799345, 13.78514932864529845373034703274, 15.64547499604769225484994205118, 16.16030001239991370174501813980, 17.40897162649728375522062915889, 17.41568124951225360560227193675, 18.50275335165158129876739462903, 18.91495198703454935137597208810, 20.63801047044014262264701318701, 20.88647769646167381974628978295, 22.15309634886302607912586360934, 22.28910821114774274241655878508, 23.26434389688325556266865988942, 23.89913624452159933516741890958
