| L(s) = 1 | − 3-s + 4-s − 2·9-s − 12-s + 16-s + 17-s + 2·23-s − 8·25-s + 5·27-s − 3·29-s − 2·36-s − 7·43-s − 48-s + 49-s − 51-s − 13·53-s + 13·61-s + 64-s + 68-s − 2·69-s + 8·75-s + 15·79-s + 81-s + 3·87-s + 2·92-s − 8·100-s − 13·101-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 2/3·9-s − 0.288·12-s + 1/4·16-s + 0.242·17-s + 0.417·23-s − 8/5·25-s + 0.962·27-s − 0.557·29-s − 1/3·36-s − 1.06·43-s − 0.144·48-s + 1/7·49-s − 0.140·51-s − 1.78·53-s + 1.66·61-s + 1/8·64-s + 0.121·68-s − 0.240·69-s + 0.923·75-s + 1.68·79-s + 1/9·81-s + 0.321·87-s + 0.208·92-s − 4/5·100-s − 1.29·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 36 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 181 T^{2} + p^{2} T^{4} \) |
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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
−8.414816504073024846209343794622, −8.099667861915831304080735542137, −7.80066944266846506076684968242, −6.99061988821415872934481417824, −6.76758674850992950959271858859, −6.17143379779245818657935420527, −5.73651952397803175602887134456, −5.30048604088420943253848877948, −4.85427199339412801567342867754, −4.00979278958666210500024117782, −3.53687163345159444280931968639, −2.83825788488785521249983737675, −2.18076971461559683274725045230, −1.33092966306737034755880833121, 0,
1.33092966306737034755880833121, 2.18076971461559683274725045230, 2.83825788488785521249983737675, 3.53687163345159444280931968639, 4.00979278958666210500024117782, 4.85427199339412801567342867754, 5.30048604088420943253848877948, 5.73651952397803175602887134456, 6.17143379779245818657935420527, 6.76758674850992950959271858859, 6.99061988821415872934481417824, 7.80066944266846506076684968242, 8.099667861915831304080735542137, 8.414816504073024846209343794622
