| L(s) = 1 | − 2·2-s + 3-s + 3·4-s − 2·6-s − 4·8-s + 9-s + 3·12-s + 5·16-s − 2·18-s − 19-s − 4·24-s − 6·25-s + 27-s + 4·29-s − 6·32-s + 3·36-s + 2·38-s − 20·41-s + 8·43-s + 5·48-s − 14·49-s + 12·50-s + 20·53-s − 2·54-s − 57-s − 8·58-s − 24·59-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 3/2·4-s − 0.816·6-s − 1.41·8-s + 1/3·9-s + 0.866·12-s + 5/4·16-s − 0.471·18-s − 0.229·19-s − 0.816·24-s − 6/5·25-s + 0.192·27-s + 0.742·29-s − 1.06·32-s + 1/2·36-s + 0.324·38-s − 3.12·41-s + 1.21·43-s + 0.721·48-s − 2·49-s + 1.69·50-s + 2.74·53-s − 0.272·54-s − 0.132·57-s − 1.05·58-s − 3.12·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 19 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.100143290691371559447654285071, −7.87114970877127535090362209983, −7.22153922721882802395635766156, −6.93381933813641935760483898363, −6.54324711643559493568388573046, −5.88087715977795198376928556594, −5.55712983394880976325928956483, −4.76383709193481805378371975040, −4.27229588770101457625379469521, −3.46272294158930356206920299963, −3.17802027729570401593965112053, −2.33039316998666771503145824429, −1.90797026202998636328713780462, −1.15871174429474862980802238748, 0,
1.15871174429474862980802238748, 1.90797026202998636328713780462, 2.33039316998666771503145824429, 3.17802027729570401593965112053, 3.46272294158930356206920299963, 4.27229588770101457625379469521, 4.76383709193481805378371975040, 5.55712983394880976325928956483, 5.88087715977795198376928556594, 6.54324711643559493568388573046, 6.93381933813641935760483898363, 7.22153922721882802395635766156, 7.87114970877127535090362209983, 8.100143290691371559447654285071
