| L(s) = 1 | − 2·3-s − 4-s − 6·5-s + 3·9-s + 2·12-s + 12·15-s − 3·16-s + 6·20-s − 12·23-s + 17·25-s − 4·27-s + 8·31-s − 3·36-s − 22·37-s − 18·45-s + 6·48-s − 2·49-s − 18·53-s − 12·59-s − 12·60-s + 7·64-s − 4·67-s + 24·69-s − 12·71-s − 34·75-s + 18·80-s + 5·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s − 2.68·5-s + 9-s + 0.577·12-s + 3.09·15-s − 3/4·16-s + 1.34·20-s − 2.50·23-s + 17/5·25-s − 0.769·27-s + 1.43·31-s − 1/2·36-s − 3.61·37-s − 2.68·45-s + 0.866·48-s − 2/7·49-s − 2.47·53-s − 1.56·59-s − 1.54·60-s + 7/8·64-s − 0.488·67-s + 2.88·69-s − 1.42·71-s − 3.92·75-s + 2.01·80-s + 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 79 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 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
−11.48613840251284076179398338363, −10.88929931871862449135005452623, −10.28328728028678558856477572353, −10.17555670712698730983302805703, −9.279137318937476680564254609479, −8.789592707949681467230495188896, −8.233106999068815384359466088835, −7.83888142451841888077368799506, −7.59634104912019193486672986525, −6.90239902824418094849867412203, −6.45942841117226412127905637665, −5.92665930083780474844521723709, −4.92362723553219248906923643921, −4.78736398400057893106922846764, −4.08874146042552895871178962658, −3.81710189102304683427755985034, −3.14712099992739018433219472729, −1.70975558990753806142579652840, 0, 0,
1.70975558990753806142579652840, 3.14712099992739018433219472729, 3.81710189102304683427755985034, 4.08874146042552895871178962658, 4.78736398400057893106922846764, 4.92362723553219248906923643921, 5.92665930083780474844521723709, 6.45942841117226412127905637665, 6.90239902824418094849867412203, 7.59634104912019193486672986525, 7.83888142451841888077368799506, 8.233106999068815384359466088835, 8.789592707949681467230495188896, 9.279137318937476680564254609479, 10.17555670712698730983302805703, 10.28328728028678558856477572353, 10.88929931871862449135005452623, 11.48613840251284076179398338363
