| L(s) = 1 | − 2·5-s + 4·13-s + 2·25-s + 8·29-s − 14·37-s + 18·41-s + 28·53-s + 20·61-s − 8·65-s + 10·73-s + 26·89-s + 26·97-s + 26·109-s − 32·113-s − 10·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.10·13-s + 2/5·25-s + 1.48·29-s − 2.30·37-s + 2.81·41-s + 3.84·53-s + 2.56·61-s − 0.992·65-s + 1.17·73-s + 2.75·89-s + 2.63·97-s + 2.49·109-s − 3.01·113-s − 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3/13·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.413833683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.413833683\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
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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
−9.098562901043967017202881254052, −9.085089984727315755970237894229, −8.497083101532657821282064117664, −8.443559099348125638355445623057, −7.86515411811517607008775054986, −7.49070219768878232430988026684, −7.03217634645471387812432730118, −6.80371590250706096906197427978, −6.20526518342110784171430923950, −5.92410276583179175432631810681, −5.25608411242825586970846982605, −5.09767789156791658935759494020, −4.31092810547757669399368250176, −4.07618261650601845342646550122, −3.52936919079648674560508386667, −3.33485695169846444418350429795, −2.32253219099474182216174591868, −2.24417931112986669965195765064, −0.982191842588274532153576588693, −0.75194921336979352191172044724,
0.75194921336979352191172044724, 0.982191842588274532153576588693, 2.24417931112986669965195765064, 2.32253219099474182216174591868, 3.33485695169846444418350429795, 3.52936919079648674560508386667, 4.07618261650601845342646550122, 4.31092810547757669399368250176, 5.09767789156791658935759494020, 5.25608411242825586970846982605, 5.92410276583179175432631810681, 6.20526518342110784171430923950, 6.80371590250706096906197427978, 7.03217634645471387812432730118, 7.49070219768878232430988026684, 7.86515411811517607008775054986, 8.443559099348125638355445623057, 8.497083101532657821282064117664, 9.085089984727315755970237894229, 9.098562901043967017202881254052
