| L(s) = 1 | + 2·2-s + 3·4-s − 4·5-s + 4·8-s − 8·10-s + 6·11-s − 8·13-s + 5·16-s − 14·17-s − 12·20-s + 12·22-s − 12·23-s + 11·25-s − 16·26-s + 6·32-s − 28·34-s + 12·37-s − 16·40-s − 14·41-s + 2·43-s + 18·44-s − 24·46-s + 5·49-s + 22·50-s − 24·52-s − 24·55-s + 7·64-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 1.78·5-s + 1.41·8-s − 2.52·10-s + 1.80·11-s − 2.21·13-s + 5/4·16-s − 3.39·17-s − 2.68·20-s + 2.55·22-s − 2.50·23-s + 11/5·25-s − 3.13·26-s + 1.06·32-s − 4.80·34-s + 1.97·37-s − 2.52·40-s − 2.18·41-s + 0.304·43-s + 2.71·44-s − 3.53·46-s + 5/7·49-s + 3.11·50-s − 3.32·52-s − 3.23·55-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11088900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11088900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 37 | $C_2$ | \( 1 - 12 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 105 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
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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
−8.359868199944863199021240091301, −7.956555329998575745283673494655, −7.41021361158273499063481624693, −7.35996393949692343753206835153, −6.76706723812939217514339957131, −6.69920253220864371515547146483, −6.16840648165814891284467185500, −5.96997046907787308616731441224, −5.09850215650327575403620761412, −4.74289753763430731219637328693, −4.47838947328881990374318444354, −4.21866124720168452169509188987, −3.85232890071935981437975915187, −3.67240067800049302071045514792, −2.67003249263391527849169126892, −2.58452729523013995160084059365, −2.03598916193688812775897295309, −1.38628598957560626554783562742, 0, 0,
1.38628598957560626554783562742, 2.03598916193688812775897295309, 2.58452729523013995160084059365, 2.67003249263391527849169126892, 3.67240067800049302071045514792, 3.85232890071935981437975915187, 4.21866124720168452169509188987, 4.47838947328881990374318444354, 4.74289753763430731219637328693, 5.09850215650327575403620761412, 5.96997046907787308616731441224, 6.16840648165814891284467185500, 6.69920253220864371515547146483, 6.76706723812939217514339957131, 7.35996393949692343753206835153, 7.41021361158273499063481624693, 7.956555329998575745283673494655, 8.359868199944863199021240091301
