| L(s) = 1 | + 32·4-s − 50·5-s − 7·9-s + 768·16-s + 1.34e3·19-s − 1.60e3·20-s + 1.87e3·25-s − 1.92e3·31-s − 224·36-s + 3.93e3·41-s + 350·45-s + 4.80e3·49-s − 1.11e4·59-s + 1.63e4·64-s − 1.73e4·71-s + 4.30e4·76-s − 3.84e4·80-s − 6.51e3·81-s − 6.73e4·95-s + 6.00e4·100-s + 2.89e4·101-s − 2.24e4·109-s + 2.92e4·121-s − 6.15e4·124-s − 6.25e4·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 2·4-s − 2·5-s − 0.0864·9-s + 3·16-s + 3.72·19-s − 4·20-s + 3·25-s − 2·31-s − 0.172·36-s + 2.34·41-s + 0.172·45-s + 2·49-s − 3.21·59-s + 4·64-s − 3.44·71-s + 7.45·76-s − 6·80-s − 0.992·81-s − 7.45·95-s + 6·100-s + 2.83·101-s − 1.88·109-s + 2·121-s − 4·124-s − 4·125-s + 6.20e−5·127-s + 5.82e−5·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.375408431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.375408431\) |
| \(L(3)\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 31 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 3 | $C_2^2$ | \( 1 + 7 T^{2} + p^{8} T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 32158 T^{2} + p^{8} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 132167 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 673 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 243838 T^{2} + p^{8} T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 3579527 T^{2} + p^{8} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 1967 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 6834793 T^{2} + p^{8} T^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 15277367 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 5593 T + p^{4} T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8687 T + p^{4} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 56512393 T^{2} + p^{8} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 5624087 T^{2} + p^{8} T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
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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
−12.10952370296662883870786108059, −12.10272958736000974717072762376, −11.39200883122234226465374941956, −11.26211087731828885138614659558, −10.77777601418047861746197108622, −10.23745495801487542936506942206, −9.383037166987002802045332827945, −8.922666784065807658255858558872, −7.954105339640689377372703284788, −7.42060745670648109722411228204, −7.39829944230310262789985880030, −7.16082308972701351259349528170, −5.89370452504852225135053691691, −5.69553931477404075917285764307, −4.70249106547682664157787478969, −3.74929387842689773065709510012, −3.12950815119666299711478881327, −2.87684022873378073923613460513, −1.47437650073585048206595801853, −0.76248597437539662967566713720,
0.76248597437539662967566713720, 1.47437650073585048206595801853, 2.87684022873378073923613460513, 3.12950815119666299711478881327, 3.74929387842689773065709510012, 4.70249106547682664157787478969, 5.69553931477404075917285764307, 5.89370452504852225135053691691, 7.16082308972701351259349528170, 7.39829944230310262789985880030, 7.42060745670648109722411228204, 7.954105339640689377372703284788, 8.922666784065807658255858558872, 9.383037166987002802045332827945, 10.23745495801487542936506942206, 10.77777601418047861746197108622, 11.26211087731828885138614659558, 11.39200883122234226465374941956, 12.10272958736000974717072762376, 12.10952370296662883870786108059
