| L(s) = 1 | − 2·4-s + 4·7-s + 4·9-s + 6·11-s + 4·16-s + 2·19-s − 10·25-s − 8·28-s − 8·36-s + 4·37-s + 4·43-s − 12·44-s − 2·49-s − 12·53-s + 16·63-s − 8·64-s − 4·76-s + 24·77-s + 16·79-s + 7·81-s + 12·83-s − 12·89-s + 28·97-s + 24·99-s + 20·100-s − 24·107-s + 16·112-s + ⋯ |
| L(s) = 1 | − 4-s + 1.51·7-s + 4/3·9-s + 1.80·11-s + 16-s + 0.458·19-s − 2·25-s − 1.51·28-s − 4/3·36-s + 0.657·37-s + 0.609·43-s − 1.80·44-s − 2/7·49-s − 1.64·53-s + 2.01·63-s − 64-s − 0.458·76-s + 2.73·77-s + 1.80·79-s + 7/9·81-s + 1.31·83-s − 1.27·89-s + 2.84·97-s + 2.41·99-s + 2·100-s − 2.32·107-s + 1.51·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 698896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 698896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.501929934\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.501929934\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 116 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−10.31709157428299308145361268172, −9.859588029598828020450038728868, −9.510105952624353843584487360990, −9.198074329083928648254700990670, −8.941467470000675594762677364772, −8.035928118087966501644620613120, −7.959451269017631778820888342513, −7.70295505094511027387440418101, −7.04961044438205057906292490561, −6.34121020170498911820054926929, −6.26719559969053777238342004880, −5.29499566010931407760516839214, −5.14334193914943671872878053081, −4.34923311324865123310854670556, −4.22931631883200751534201738319, −3.82902123107670144269340963908, −3.14202125826621676752835184493, −1.85275428010270077320620152888, −1.64021477324757180584491426343, −0.892755678421709968553201593444,
0.892755678421709968553201593444, 1.64021477324757180584491426343, 1.85275428010270077320620152888, 3.14202125826621676752835184493, 3.82902123107670144269340963908, 4.22931631883200751534201738319, 4.34923311324865123310854670556, 5.14334193914943671872878053081, 5.29499566010931407760516839214, 6.26719559969053777238342004880, 6.34121020170498911820054926929, 7.04961044438205057906292490561, 7.70295505094511027387440418101, 7.959451269017631778820888342513, 8.035928118087966501644620613120, 8.941467470000675594762677364772, 9.198074329083928648254700990670, 9.510105952624353843584487360990, 9.859588029598828020450038728868, 10.31709157428299308145361268172
