| L(s) = 1 | + 2·2-s − 4-s − 8·8-s − 6·9-s + 8·11-s − 7·16-s − 12·18-s + 16·22-s + 16·23-s − 10·25-s + 4·29-s + 14·32-s + 6·36-s − 12·37-s − 24·43-s − 8·44-s + 32·46-s − 20·50-s − 20·53-s + 8·58-s + 35·64-s + 8·67-s + 32·71-s + 48·72-s − 24·74-s + 16·79-s + 27·81-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s − 2.82·8-s − 2·9-s + 2.41·11-s − 7/4·16-s − 2.82·18-s + 3.41·22-s + 3.33·23-s − 2·25-s + 0.742·29-s + 2.47·32-s + 36-s − 1.97·37-s − 3.65·43-s − 1.20·44-s + 4.71·46-s − 2.82·50-s − 2.74·53-s + 1.05·58-s + 35/8·64-s + 0.977·67-s + 3.79·71-s + 5.65·72-s − 2.78·74-s + 1.80·79-s + 3·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9344235377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9344235377\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−18.93330835276049829789695936351, −17.58953823246936007643954304401, −17.58953823246936007643954304401, −16.91414841722080276990447443364, −16.91414841722080276990447443364, −15.25649719028727745654998030256, −15.25649719028727745654998030256, −14.36789003254179224985767994212, −14.36789003254179224985767994212, −13.55829126515041538892762617601, −13.55829126515041538892762617601, −12.27943344760252552732176729088, −12.27943344760252552732176729088, −11.30850265031875128052013945557, −11.30850265031875128052013945557, −9.489525085039792097687572288594, −9.489525085039792097687572288594, −8.498120181782134288657898916471, −8.498120181782134288657898916471, −6.47803659589426412986704519993, −6.47803659589426412986704519993, −5.08673463814783736975452386140, −5.08673463814783736975452386140, −3.45773984941604093394269964405, −3.45773984941604093394269964405,
3.45773984941604093394269964405, 3.45773984941604093394269964405, 5.08673463814783736975452386140, 5.08673463814783736975452386140, 6.47803659589426412986704519993, 6.47803659589426412986704519993, 8.498120181782134288657898916471, 8.498120181782134288657898916471, 9.489525085039792097687572288594, 9.489525085039792097687572288594, 11.30850265031875128052013945557, 11.30850265031875128052013945557, 12.27943344760252552732176729088, 12.27943344760252552732176729088, 13.55829126515041538892762617601, 13.55829126515041538892762617601, 14.36789003254179224985767994212, 14.36789003254179224985767994212, 15.25649719028727745654998030256, 15.25649719028727745654998030256, 16.91414841722080276990447443364, 16.91414841722080276990447443364, 17.58953823246936007643954304401, 17.58953823246936007643954304401, 18.93330835276049829789695936351
