| L(s) = 1 | − 3-s + 9-s − 11-s − 8·17-s − 6·25-s − 27-s + 33-s − 12·37-s − 10·49-s + 8·51-s + 8·67-s + 6·75-s + 81-s + 32·83-s − 4·97-s − 99-s + 16·101-s − 24·103-s + 32·107-s + 12·111-s + 121-s + 127-s + 131-s + 137-s + 139-s + 10·147-s + 149-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.301·11-s − 1.94·17-s − 6/5·25-s − 0.192·27-s + 0.174·33-s − 1.97·37-s − 1.42·49-s + 1.12·51-s + 0.977·67-s + 0.692·75-s + 1/9·81-s + 3.51·83-s − 0.406·97-s − 0.100·99-s + 1.59·101-s − 2.36·103-s + 3.09·107-s + 1.13·111-s + 1/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.824·147-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 574992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8340465365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8340465365\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.363783361006180122238656454062, −8.017674089261979349140094913788, −7.49718153606969186728697393667, −7.00719281751835795853753965763, −6.41893627610765861044252902088, −6.41512539021783853545914554633, −5.62860857478680063008223727075, −5.12414628938185870335019819699, −4.78818744575862870088978876704, −4.18020563951979381603420573914, −3.67380578777293619845336752383, −3.05194549653743224577997918283, −2.07022010244043316224023560102, −1.86785039249087723872477729694, −0.46511289287587154532633948638,
0.46511289287587154532633948638, 1.86785039249087723872477729694, 2.07022010244043316224023560102, 3.05194549653743224577997918283, 3.67380578777293619845336752383, 4.18020563951979381603420573914, 4.78818744575862870088978876704, 5.12414628938185870335019819699, 5.62860857478680063008223727075, 6.41512539021783853545914554633, 6.41893627610765861044252902088, 7.00719281751835795853753965763, 7.49718153606969186728697393667, 8.017674089261979349140094913788, 8.363783361006180122238656454062
