| L(s) = 1 | + 3·5-s − 5·7-s + 9·11-s + 3·17-s + 3·19-s − 9·23-s + 5·25-s − 15·35-s − 7·37-s + 6·41-s − 4·43-s − 6·47-s + 18·49-s + 9·53-s + 27·55-s + 6·59-s + 10·67-s − 21·73-s − 45·77-s − 2·79-s − 12·83-s + 9·85-s − 9·89-s + 9·95-s − 12·97-s − 9·101-s + 9·103-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.88·7-s + 2.71·11-s + 0.727·17-s + 0.688·19-s − 1.87·23-s + 25-s − 2.53·35-s − 1.15·37-s + 0.937·41-s − 0.609·43-s − 0.875·47-s + 18/7·49-s + 1.23·53-s + 3.64·55-s + 0.781·59-s + 1.22·67-s − 2.45·73-s − 5.12·77-s − 0.225·79-s − 1.31·83-s + 0.976·85-s − 0.953·89-s + 0.923·95-s − 1.21·97-s − 0.895·101-s + 0.886·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.883200848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.883200848\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 80 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 21 T + 220 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 9 T - 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 12 T + 145 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−9.475314982450667672753006974768, −8.898132391120279397051382019732, −8.580876003084758523792441700443, −8.327463111434134468119983488325, −7.38370751088466463750329083236, −7.13705027816024539978647781625, −6.88208097560779866532744590623, −6.26190093283656194515921359978, −6.25014217934046606642899755385, −5.77333479929807323163705444281, −5.58479628492002043752380575277, −4.88029087446747298954614810207, −4.11048280712447170570205183799, −3.89568658242517958843302995160, −3.55924220400289672328005687341, −2.98081878767608433270158170072, −2.51783444075391843577383600379, −1.69098205460422282567412747546, −1.45885502087729930332374304246, −0.59934417130682844962228825563,
0.59934417130682844962228825563, 1.45885502087729930332374304246, 1.69098205460422282567412747546, 2.51783444075391843577383600379, 2.98081878767608433270158170072, 3.55924220400289672328005687341, 3.89568658242517958843302995160, 4.11048280712447170570205183799, 4.88029087446747298954614810207, 5.58479628492002043752380575277, 5.77333479929807323163705444281, 6.25014217934046606642899755385, 6.26190093283656194515921359978, 6.88208097560779866532744590623, 7.13705027816024539978647781625, 7.38370751088466463750329083236, 8.327463111434134468119983488325, 8.580876003084758523792441700443, 8.898132391120279397051382019732, 9.475314982450667672753006974768
