| L(s) = 1 | − 2·3-s + 3·9-s − 6·13-s − 12·17-s − 25-s − 4·27-s + 4·29-s + 12·39-s + 8·43-s + 10·49-s + 24·51-s − 28·53-s − 12·61-s + 2·75-s − 32·79-s + 5·81-s − 8·87-s − 36·101-s − 16·103-s − 16·107-s − 12·113-s − 18·117-s + 22·121-s + 127-s − 16·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s − 1.66·13-s − 2.91·17-s − 1/5·25-s − 0.769·27-s + 0.742·29-s + 1.92·39-s + 1.21·43-s + 10/7·49-s + 3.36·51-s − 3.84·53-s − 1.53·61-s + 0.230·75-s − 3.60·79-s + 5/9·81-s − 0.857·87-s − 3.58·101-s − 1.57·103-s − 1.54·107-s − 1.12·113-s − 1.66·117-s + 2·121-s + 0.0887·127-s − 1.40·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.232748252366323224155688346354, −9.044414498610368811653042126421, −8.371822554715386064253988057727, −8.090611164197294015974882354072, −7.32504657139876138557960073336, −7.30006299249593301398255106866, −6.63565682695802801519112379455, −6.57420374467357792986395884776, −5.94134093902010745882085885364, −5.62328757595858029594718124014, −4.92815088388699293508799438723, −4.72213467900278094244266697734, −4.22746920114492678382378393552, −4.11752299861170025040273496022, −2.80985517862297669190151658510, −2.76494825941652098652757009675, −1.96531439327367176579886998544, −1.36860532639644266450366422905, 0, 0,
1.36860532639644266450366422905, 1.96531439327367176579886998544, 2.76494825941652098652757009675, 2.80985517862297669190151658510, 4.11752299861170025040273496022, 4.22746920114492678382378393552, 4.72213467900278094244266697734, 4.92815088388699293508799438723, 5.62328757595858029594718124014, 5.94134093902010745882085885364, 6.57420374467357792986395884776, 6.63565682695802801519112379455, 7.30006299249593301398255106866, 7.32504657139876138557960073336, 8.090611164197294015974882354072, 8.371822554715386064253988057727, 9.044414498610368811653042126421, 9.232748252366323224155688346354
