| L(s) = 1 | − 3·4-s − 6·9-s − 8·11-s + 5·16-s + 16·23-s − 10·25-s + 18·36-s − 12·43-s + 24·44-s − 7·49-s + 20·53-s − 3·64-s − 8·67-s − 16·79-s + 27·81-s − 48·92-s + 48·99-s + 30·100-s + 40·107-s + 36·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s − 30·144-s + 149-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 2·9-s − 2.41·11-s + 5/4·16-s + 3.33·23-s − 2·25-s + 3·36-s − 1.82·43-s + 3.61·44-s − 49-s + 2.74·53-s − 3/8·64-s − 0.977·67-s − 1.80·79-s + 3·81-s − 5.00·92-s + 4.82·99-s + 3·100-s + 3.86·107-s + 3.44·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5/2·144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90601 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90601 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3770149412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3770149412\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_2$ | \( 1 + p T^{2} \) |
| 43 | $C_2$ | \( 1 + 12 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−12.77804918155507843432324714169, −11.53619817530416009706665569271, −11.11061969916087762209884752293, −10.41653003036951018854115807397, −10.17432549174581265691449505266, −9.608824368965476585695529566832, −8.917069912745143464449274409598, −8.557303031440003425560599675659, −8.542173983774536735452442640985, −7.58112690918246378526733933846, −7.53513597869528252859719214749, −6.49133150407914219972922655312, −5.63949344585099693288013133217, −5.43511014783711324462565459686, −5.03324083434124411378565038941, −4.52921798040596027546976096168, −3.38483437827663718532385256099, −3.08172768293520709156579267877, −2.32529114495696054308565962392, −0.44324083827136314704741050533,
0.44324083827136314704741050533, 2.32529114495696054308565962392, 3.08172768293520709156579267877, 3.38483437827663718532385256099, 4.52921798040596027546976096168, 5.03324083434124411378565038941, 5.43511014783711324462565459686, 5.63949344585099693288013133217, 6.49133150407914219972922655312, 7.53513597869528252859719214749, 7.58112690918246378526733933846, 8.542173983774536735452442640985, 8.557303031440003425560599675659, 8.917069912745143464449274409598, 9.608824368965476585695529566832, 10.17432549174581265691449505266, 10.41653003036951018854115807397, 11.11061969916087762209884752293, 11.53619817530416009706665569271, 12.77804918155507843432324714169
