| L(s) = 1 | + 8·4-s − 10·5-s + 13·9-s + 48·16-s − 14·19-s − 80·20-s + 75·25-s + 62·31-s + 104·36-s − 146·41-s − 130·45-s + 98·49-s − 74·59-s + 256·64-s + 274·71-s − 112·76-s − 480·80-s + 88·81-s + 140·95-s + 600·100-s + 154·101-s − 374·109-s + 242·121-s + 496·124-s − 500·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 2·4-s − 2·5-s + 13/9·9-s + 3·16-s − 0.736·19-s − 4·20-s + 3·25-s + 2·31-s + 26/9·36-s − 3.56·41-s − 2.88·45-s + 2·49-s − 1.25·59-s + 4·64-s + 3.85·71-s − 1.47·76-s − 6·80-s + 1.08·81-s + 1.47·95-s + 6·100-s + 1.52·101-s − 3.43·109-s + 2·121-s + 4·124-s − 4·125-s + 0.00787·127-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.374966405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.374966405\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 31 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 13 T^{2} + p^{4} T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 158 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 547 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 562 T^{2} + p^{4} T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2707 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 73 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 53 T^{2} + p^{4} T^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 5573 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 37 T + p^{2} T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 137 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 533 T^{2} + p^{4} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 10027 T^{2} + p^{4} T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
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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
−12.60658925002419500097703861921, −12.23330202683516590332915226372, −11.97142515661368660037100047814, −11.61999703409265825674446992188, −10.97686295420063541958134314676, −10.53306246679009811197752946687, −10.30972577782085880859362828480, −9.567776166775899902141785583790, −8.319977614499004022906743925798, −8.294976018017822736504687884204, −7.64464915890197745923974163017, −6.99627985417353562985457041374, −6.83520120103236888398668866800, −6.36086296049252241924958256501, −5.19106524458049648605708515887, −4.46915178273386682515378181205, −3.72081582542140240368648837292, −3.18440886171714944092537616672, −2.17439898010004839334309970546, −1.05935429144057492014198067969,
1.05935429144057492014198067969, 2.17439898010004839334309970546, 3.18440886171714944092537616672, 3.72081582542140240368648837292, 4.46915178273386682515378181205, 5.19106524458049648605708515887, 6.36086296049252241924958256501, 6.83520120103236888398668866800, 6.99627985417353562985457041374, 7.64464915890197745923974163017, 8.294976018017822736504687884204, 8.319977614499004022906743925798, 9.567776166775899902141785583790, 10.30972577782085880859362828480, 10.53306246679009811197752946687, 10.97686295420063541958134314676, 11.61999703409265825674446992188, 11.97142515661368660037100047814, 12.23330202683516590332915226372, 12.60658925002419500097703861921
