| L(s) = 1 | + 2·3-s − 4·5-s − 2·7-s + 3·9-s − 8·15-s − 4·17-s + 2·19-s − 4·21-s + 4·25-s + 4·27-s + 12·29-s + 12·31-s + 8·35-s − 12·37-s − 12·41-s − 12·43-s − 12·45-s − 24·47-s + 3·49-s − 8·51-s + 4·53-s + 4·57-s − 16·59-s + 4·61-s − 6·63-s − 24·67-s − 12·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s − 0.755·7-s + 9-s − 2.06·15-s − 0.970·17-s + 0.458·19-s − 0.872·21-s + 4/5·25-s + 0.769·27-s + 2.22·29-s + 2.15·31-s + 1.35·35-s − 1.97·37-s − 1.87·41-s − 1.82·43-s − 1.78·45-s − 3.50·47-s + 3/7·49-s − 1.12·51-s + 0.549·53-s + 0.529·57-s − 2.08·59-s + 0.512·61-s − 0.755·63-s − 2.93·67-s − 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10188864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10188864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 12 T + 76 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 24 T + 236 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 60 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 24 T + 270 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 212 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 28 T + 366 T^{2} + 28 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 198 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−8.462399436032068907590945347015, −8.274935678345561594433718390332, −7.903796292839556892021560512037, −7.37413780590136710789175356611, −6.98997765969264461242392868366, −6.83477816890128458220995817469, −6.23352844012666151409034838664, −6.23143646501705730214391949789, −5.03506044716846449582163155312, −4.94701896220361361764655710890, −4.47071256086318044548912069066, −4.20547437352001516235749198810, −3.44354079713731384784731494977, −3.40258510178812708964369422359, −2.93877897115368570253503994234, −2.66814985854442093997786777427, −1.61911530681874039405030078714, −1.40567775873301748409140799060, 0, 0,
1.40567775873301748409140799060, 1.61911530681874039405030078714, 2.66814985854442093997786777427, 2.93877897115368570253503994234, 3.40258510178812708964369422359, 3.44354079713731384784731494977, 4.20547437352001516235749198810, 4.47071256086318044548912069066, 4.94701896220361361764655710890, 5.03506044716846449582163155312, 6.23143646501705730214391949789, 6.23352844012666151409034838664, 6.83477816890128458220995817469, 6.98997765969264461242392868366, 7.37413780590136710789175356611, 7.903796292839556892021560512037, 8.274935678345561594433718390332, 8.462399436032068907590945347015
