| L(s) = 1 | + 4·3-s + 4-s − 6·5-s + 4·7-s + 6·9-s + 4·12-s − 24·15-s − 3·16-s − 6·20-s + 16·21-s + 17·25-s − 4·27-s + 4·28-s − 6·29-s + 20·31-s − 24·35-s + 6·36-s − 4·43-s − 36·45-s − 12·48-s − 2·49-s − 24·60-s − 14·61-s + 24·63-s − 7·64-s − 12·71-s − 10·73-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1/2·4-s − 2.68·5-s + 1.51·7-s + 2·9-s + 1.15·12-s − 6.19·15-s − 3/4·16-s − 1.34·20-s + 3.49·21-s + 17/5·25-s − 0.769·27-s + 0.755·28-s − 1.11·29-s + 3.59·31-s − 4.05·35-s + 36-s − 0.609·43-s − 5.36·45-s − 1.73·48-s − 2/7·49-s − 3.09·60-s − 1.79·61-s + 3.02·63-s − 7/8·64-s − 1.42·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11881 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11881 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.620964690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.620964690\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 109 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 43 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 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
−14.20232454270548124407005727142, −13.48164518089194052947074051837, −13.25157073545991747739357669521, −12.03647411432218026980308105710, −11.70293629711369096868126013036, −11.67202681492915810561346707467, −10.99815185705015503633140995728, −10.33010835072647844936150469193, −9.299859148714540862128462661432, −8.835405866348020997702142121052, −8.235216332727073392467013788593, −8.029592951776005817274904617370, −7.72109027701982789626761439014, −7.29650716703238363441116451472, −6.27521693860836324238409605199, −4.60760839967832260992047870441, −4.50757905019875647891694531806, −3.49996828010670678507555267190, −3.07576554973497310823571976342, −2.06835638094502799927523372775,
2.06835638094502799927523372775, 3.07576554973497310823571976342, 3.49996828010670678507555267190, 4.50757905019875647891694531806, 4.60760839967832260992047870441, 6.27521693860836324238409605199, 7.29650716703238363441116451472, 7.72109027701982789626761439014, 8.029592951776005817274904617370, 8.235216332727073392467013788593, 8.835405866348020997702142121052, 9.299859148714540862128462661432, 10.33010835072647844936150469193, 10.99815185705015503633140995728, 11.67202681492915810561346707467, 11.70293629711369096868126013036, 12.03647411432218026980308105710, 13.25157073545991747739357669521, 13.48164518089194052947074051837, 14.20232454270548124407005727142
