| L(s) = 1 | − 4-s − 4·13-s − 3·16-s + 8·19-s + 2·25-s + 8·31-s + 4·37-s − 8·43-s + 4·52-s + 20·61-s + 7·64-s − 8·67-s − 28·73-s − 8·76-s + 16·79-s − 28·97-s − 2·100-s + 8·103-s + 4·109-s − 10·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + 149-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.10·13-s − 3/4·16-s + 1.83·19-s + 2/5·25-s + 1.43·31-s + 0.657·37-s − 1.21·43-s + 0.554·52-s + 2.56·61-s + 7/8·64-s − 0.977·67-s − 3.27·73-s − 0.917·76-s + 1.80·79-s − 2.84·97-s − 1/5·100-s + 0.788·103-s + 0.383·109-s − 0.909·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.328·148-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.291338071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.291338071\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−11.48393794420244135371546013396, −10.95658054654902885523846772136, −10.17188386146879038957946866480, −10.06144849154688278201195905973, −9.508923167256671296794728048315, −9.268646300567467243411926807400, −8.588054341470238711310997797416, −8.225844865232840812413709801409, −7.69810934092326468321480626486, −7.11368080298575350370680898060, −6.86728429350633734444790732393, −6.21520433376882407027295525295, −5.34195569456028256586443128798, −5.29179036278425807274949680025, −4.45556336501987772086854948897, −4.23077073410533406981583562252, −3.13108407536332909137163043288, −2.85720623146223462659941831225, −1.88630160201182295709583291815, −0.74095505323005833418238937755,
0.74095505323005833418238937755, 1.88630160201182295709583291815, 2.85720623146223462659941831225, 3.13108407536332909137163043288, 4.23077073410533406981583562252, 4.45556336501987772086854948897, 5.29179036278425807274949680025, 5.34195569456028256586443128798, 6.21520433376882407027295525295, 6.86728429350633734444790732393, 7.11368080298575350370680898060, 7.69810934092326468321480626486, 8.225844865232840812413709801409, 8.588054341470238711310997797416, 9.268646300567467243411926807400, 9.508923167256671296794728048315, 10.06144849154688278201195905973, 10.17188386146879038957946866480, 10.95658054654902885523846772136, 11.48393794420244135371546013396
