| L(s) = 1 | − 4·7-s − 8·25-s + 8·37-s − 16·43-s − 2·49-s − 32·67-s − 40·79-s − 9·81-s − 40·109-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s + 32·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 8/5·25-s + 1.31·37-s − 2.43·43-s − 2/7·49-s − 3.90·67-s − 4.50·79-s − 81-s − 3.83·109-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.07·169-s + 0.0760·173-s + 2.41·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4704624008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4704624008\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.89693733821419862029344013445, −6.77130199034238236588927035027, −6.35644266529429407330842607946, −6.09322676392984717018518930736, −6.02753901704502443368209942138, −5.93368740113836409080991762487, −5.50825788420763745272891317105, −5.46636012062636927595826995859, −5.22651224072673337728531477348, −4.66513689231864600778735069544, −4.57104469491077490045295172069, −4.34475638788794329020820104505, −4.24600916170197955468920684760, −3.84392691859893675269185208098, −3.68112011416872061149464306207, −3.15061004507011078291461059817, −3.02687637249399231725788108467, −3.00212806464306532900267735890, −2.81233117704625371871734739836, −2.24707519718873489754290335895, −1.78850203735628051610877710490, −1.60127924413133821402085607501, −1.48102845134630676094188541614, −0.63951071247231076208939433919, −0.18210282536723180763365338167,
0.18210282536723180763365338167, 0.63951071247231076208939433919, 1.48102845134630676094188541614, 1.60127924413133821402085607501, 1.78850203735628051610877710490, 2.24707519718873489754290335895, 2.81233117704625371871734739836, 3.00212806464306532900267735890, 3.02687637249399231725788108467, 3.15061004507011078291461059817, 3.68112011416872061149464306207, 3.84392691859893675269185208098, 4.24600916170197955468920684760, 4.34475638788794329020820104505, 4.57104469491077490045295172069, 4.66513689231864600778735069544, 5.22651224072673337728531477348, 5.46636012062636927595826995859, 5.50825788420763745272891317105, 5.93368740113836409080991762487, 6.02753901704502443368209942138, 6.09322676392984717018518930736, 6.35644266529429407330842607946, 6.77130199034238236588927035027, 6.89693733821419862029344013445
