| L(s) = 1 | + 2-s + 4·3-s − 4-s + 4·6-s − 3·8-s + 6·9-s − 4·12-s − 16-s − 10·17-s + 6·18-s + 12·19-s − 12·24-s − 9·25-s − 4·27-s + 5·32-s − 10·34-s − 6·36-s + 12·38-s − 10·41-s − 4·48-s − 10·49-s − 9·50-s − 40·51-s − 4·54-s + 48·57-s + 16·59-s + 7·64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 2.30·3-s − 1/2·4-s + 1.63·6-s − 1.06·8-s + 2·9-s − 1.15·12-s − 1/4·16-s − 2.42·17-s + 1.41·18-s + 2.75·19-s − 2.44·24-s − 9/5·25-s − 0.769·27-s + 0.883·32-s − 1.71·34-s − 36-s + 1.94·38-s − 1.56·41-s − 0.577·48-s − 1.42·49-s − 1.27·50-s − 5.60·51-s − 0.544·54-s + 6.35·57-s + 2.08·59-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 937024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 937024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
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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
−8.136008797220280319419228845171, −7.67392442093299937851307415272, −7.16194958523538100095638041627, −6.72221402891887535126565651102, −6.13889121971951459563759545037, −5.38415572352897256696127903868, −5.31187507267702842666300711205, −4.54500739067419388273186866616, −3.84177902362047353880486406829, −3.80769101471314331576105678498, −3.14661394972730814130243727198, −2.75274084463573300677827364197, −2.24761395244869231865045671648, −1.57662902890973747110699883088, 0,
1.57662902890973747110699883088, 2.24761395244869231865045671648, 2.75274084463573300677827364197, 3.14661394972730814130243727198, 3.80769101471314331576105678498, 3.84177902362047353880486406829, 4.54500739067419388273186866616, 5.31187507267702842666300711205, 5.38415572352897256696127903868, 6.13889121971951459563759545037, 6.72221402891887535126565651102, 7.16194958523538100095638041627, 7.67392442093299937851307415272, 8.136008797220280319419228845171
