| L(s) = 1 | − 8·5-s − 2·7-s − 11·9-s + 28·11-s + 46·17-s + 20·19-s − 2·23-s − 2·25-s + 16·35-s + 136·43-s + 88·45-s + 52·47-s − 95·49-s − 224·55-s − 80·61-s + 22·63-s − 14·73-s − 56·77-s + 40·81-s + 64·83-s − 368·85-s − 160·95-s − 308·99-s + 28·101-s + 16·115-s − 92·119-s + 346·121-s + ⋯ |
| L(s) = 1 | − 8/5·5-s − 2/7·7-s − 1.22·9-s + 2.54·11-s + 2.70·17-s + 1.05·19-s − 0.0869·23-s − 0.0799·25-s + 0.457·35-s + 3.16·43-s + 1.95·45-s + 1.10·47-s − 1.93·49-s − 4.07·55-s − 1.31·61-s + 0.349·63-s − 0.191·73-s − 0.727·77-s + 0.493·81-s + 0.771·83-s − 4.32·85-s − 1.68·95-s − 3.11·99-s + 0.277·101-s + 0.139·115-s − 0.773·119-s + 2.85·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.133116023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.133116023\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 20 T + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 11 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 77 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 23 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 23 p T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 878 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 1694 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2318 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 68 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 907 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6701 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 8717 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 9038 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 3086 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 862 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 9422 T^{2} + p^{4} T^{4} \) |
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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
−14.73792726753158658111641539903, −14.06909735597060663396202096686, −13.83112550727806238278890888257, −12.46954979555055655138545642500, −12.19540344222557387263325395837, −11.86287284820905195503533809962, −11.46964466416911543284905783279, −10.95054298012902263930878320015, −9.928181602485814240881104085420, −9.353801653649304380165595301930, −8.981873131895439639487253273588, −7.966893195597801610108853043856, −7.76494116857753110274493080363, −7.06561349234113979811058530109, −6.01662473603633316259496512174, −5.69753480996872523322426781716, −4.35706435694904317520744705856, −3.62646101573266471213030002533, −3.23531562519120511534305102665, −1.02410756581701884745227159159,
1.02410756581701884745227159159, 3.23531562519120511534305102665, 3.62646101573266471213030002533, 4.35706435694904317520744705856, 5.69753480996872523322426781716, 6.01662473603633316259496512174, 7.06561349234113979811058530109, 7.76494116857753110274493080363, 7.966893195597801610108853043856, 8.981873131895439639487253273588, 9.353801653649304380165595301930, 9.928181602485814240881104085420, 10.95054298012902263930878320015, 11.46964466416911543284905783279, 11.86287284820905195503533809962, 12.19540344222557387263325395837, 12.46954979555055655138545642500, 13.83112550727806238278890888257, 14.06909735597060663396202096686, 14.73792726753158658111641539903
