| L(s) = 1 | − 2·2-s + 3·4-s + 2·7-s − 4·8-s + 6·11-s − 4·14-s + 5·16-s − 12·22-s + 12·23-s − 10·25-s + 6·28-s − 12·29-s − 6·32-s − 8·37-s − 2·43-s + 18·44-s − 24·46-s − 3·49-s + 20·50-s − 24·53-s − 8·56-s + 24·58-s + 7·64-s + 10·67-s + 24·71-s + 16·74-s + 12·77-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.755·7-s − 1.41·8-s + 1.80·11-s − 1.06·14-s + 5/4·16-s − 2.55·22-s + 2.50·23-s − 2·25-s + 1.13·28-s − 2.22·29-s − 1.06·32-s − 1.31·37-s − 0.304·43-s + 2.71·44-s − 3.53·46-s − 3/7·49-s + 2.82·50-s − 3.29·53-s − 1.06·56-s + 3.15·58-s + 7/8·64-s + 1.22·67-s + 2.84·71-s + 1.85·74-s + 1.36·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−7.80670854475898716623947697687, −7.57486075433264375877702850521, −6.83856700403258930718083224666, −6.68619713403218599725876916726, −6.35782135459828133998327355434, −5.48525873198846016080664036461, −5.34383976963704470462884839275, −4.67165130826644550690102729752, −3.84523654364808887700815920458, −3.65076060485342096132488694617, −2.98455636915823987690855594247, −2.07379971708251484770168742942, −1.60693251106342657124988453680, −1.22338111669750545882567189127, 0,
1.22338111669750545882567189127, 1.60693251106342657124988453680, 2.07379971708251484770168742942, 2.98455636915823987690855594247, 3.65076060485342096132488694617, 3.84523654364808887700815920458, 4.67165130826644550690102729752, 5.34383976963704470462884839275, 5.48525873198846016080664036461, 6.35782135459828133998327355434, 6.68619713403218599725876916726, 6.83856700403258930718083224666, 7.57486075433264375877702850521, 7.80670854475898716623947697687
