| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 4·6-s − 4·8-s + 3·9-s + 2·11-s − 6·12-s + 5·16-s + 8·17-s − 6·18-s + 8·19-s − 4·22-s + 4·23-s + 8·24-s − 10·25-s − 4·27-s + 8·31-s − 6·32-s − 4·33-s − 16·34-s + 9·36-s − 4·37-s − 16·38-s + 8·41-s − 20·43-s + 6·44-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 1.41·8-s + 9-s + 0.603·11-s − 1.73·12-s + 5/4·16-s + 1.94·17-s − 1.41·18-s + 1.83·19-s − 0.852·22-s + 0.834·23-s + 1.63·24-s − 2·25-s − 0.769·27-s + 1.43·31-s − 1.06·32-s − 0.696·33-s − 2.74·34-s + 3/2·36-s − 0.657·37-s − 2.59·38-s + 1.24·41-s − 3.04·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10458756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10458756 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.200527873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.200527873\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 28 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 92 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 120 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 180 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 192 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
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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
−8.766960427947677118543865671761, −8.466338514208246352078610631754, −8.068463522041712263902291660401, −7.72148591126973706192010012839, −7.37667243631191563309784703290, −7.13024924534784689471564151413, −6.54103318366231672179782714745, −6.44899630446655505574090735130, −5.71197894861625493559687698678, −5.68576150066804518200048010910, −5.06334681038250510709292966591, −4.98107000346534109081349952471, −3.93069148420049297885244480397, −3.82779244373936436383246083855, −3.04439911837506710661099257336, −2.94183393649034862619246894418, −1.77501293350764547644740597474, −1.68477810551715713715733220735, −0.78709164801638357644427046107, −0.70901587570065549658501521432,
0.70901587570065549658501521432, 0.78709164801638357644427046107, 1.68477810551715713715733220735, 1.77501293350764547644740597474, 2.94183393649034862619246894418, 3.04439911837506710661099257336, 3.82779244373936436383246083855, 3.93069148420049297885244480397, 4.98107000346534109081349952471, 5.06334681038250510709292966591, 5.68576150066804518200048010910, 5.71197894861625493559687698678, 6.44899630446655505574090735130, 6.54103318366231672179782714745, 7.13024924534784689471564151413, 7.37667243631191563309784703290, 7.72148591126973706192010012839, 8.068463522041712263902291660401, 8.466338514208246352078610631754, 8.766960427947677118543865671761
