| L(s) = 1 | + 2·7-s − 2·13-s − 4·16-s − 19-s − 5·25-s + 14·31-s − 12·37-s − 6·43-s − 7·49-s + 14·61-s + 16·67-s + 22·73-s + 16·79-s − 4·91-s − 6·97-s + 6·103-s − 20·109-s − 8·112-s + 7·121-s + 127-s + 131-s − 2·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.554·13-s − 16-s − 0.229·19-s − 25-s + 2.51·31-s − 1.97·37-s − 0.914·43-s − 49-s + 1.79·61-s + 1.95·67-s + 2.57·73-s + 1.80·79-s − 0.419·91-s − 0.609·97-s + 0.591·103-s − 1.91·109-s − 0.755·112-s + 7/11·121-s + 0.0887·127-s + 0.0873·131-s − 0.173·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 555579 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555579 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.714089374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.714089374\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 37 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.378055748517389919201916326891, −8.075068509879124179418625915213, −7.72819441190072061251667077294, −6.86596307298395218114703112001, −6.73603975463353222731211952493, −6.38112736969342646540994972004, −5.51187602337893547655286333658, −5.08706446260923011781257033891, −4.82610782827509455945957151913, −4.17667762547121136399357394698, −3.68017594462658460963184754975, −2.96406184154705961405374883536, −2.18289942719005744119345832946, −1.86770081326337208261340633823, −0.66836180789805474492001627331,
0.66836180789805474492001627331, 1.86770081326337208261340633823, 2.18289942719005744119345832946, 2.96406184154705961405374883536, 3.68017594462658460963184754975, 4.17667762547121136399357394698, 4.82610782827509455945957151913, 5.08706446260923011781257033891, 5.51187602337893547655286333658, 6.38112736969342646540994972004, 6.73603975463353222731211952493, 6.86596307298395218114703112001, 7.72819441190072061251667077294, 8.075068509879124179418625915213, 8.378055748517389919201916326891
