L(s) = 1 | + 2·3-s + 4·5-s + 2·7-s + 2·9-s − 6·13-s + 8·15-s − 6·17-s + 12·19-s + 4·21-s + 6·23-s + 11·25-s + 6·27-s + 8·35-s − 6·37-s − 12·39-s − 12·41-s + 6·43-s + 8·45-s − 18·47-s + 2·49-s − 12·51-s − 10·53-s + 24·57-s + 20·59-s + 24·61-s + 4·63-s − 24·65-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1.78·5-s + 0.755·7-s + 2/3·9-s − 1.66·13-s + 2.06·15-s − 1.45·17-s + 2.75·19-s + 0.872·21-s + 1.25·23-s + 11/5·25-s + 1.15·27-s + 1.35·35-s − 0.986·37-s − 1.92·39-s − 1.87·41-s + 0.914·43-s + 1.19·45-s − 2.62·47-s + 2/7·49-s − 1.68·51-s − 1.37·53-s + 3.17·57-s + 2.60·59-s + 3.07·61-s + 0.503·63-s − 2.97·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.697576054\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.697576054\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−9.931579486148577430564886739013, −9.449067346066880816214241498702, −9.088594341794882929311904071148, −8.784331494691987531536829862137, −8.185102972063888876006281941771, −8.159521276332962987214068197197, −7.16164911666349716168187676968, −7.04774707519297824865275107088, −6.87628114886777539893296168708, −6.22676127035704213898525979373, −5.28493979091711342147672728824, −5.24322858877709291828523680007, −5.01182897657812066044321454603, −4.49854930871908237551528334281, −3.40769459175002424497657637800, −3.26429591846560189888122359870, −2.43378473758342156522800465222, −2.33519711984303089599648813004, −1.65489594796210385618329486449, −0.986084598461864850013634427197,
0.986084598461864850013634427197, 1.65489594796210385618329486449, 2.33519711984303089599648813004, 2.43378473758342156522800465222, 3.26429591846560189888122359870, 3.40769459175002424497657637800, 4.49854930871908237551528334281, 5.01182897657812066044321454603, 5.24322858877709291828523680007, 5.28493979091711342147672728824, 6.22676127035704213898525979373, 6.87628114886777539893296168708, 7.04774707519297824865275107088, 7.16164911666349716168187676968, 8.159521276332962987214068197197, 8.185102972063888876006281941771, 8.784331494691987531536829862137, 9.088594341794882929311904071148, 9.449067346066880816214241498702, 9.931579486148577430564886739013