| L(s) = 1 | − 6·9-s + 8·13-s − 24·37-s + 20·41-s + 14·49-s − 8·53-s + 27·81-s + 20·89-s − 48·117-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 2·9-s + 2.21·13-s − 3.94·37-s + 3.12·41-s + 2·49-s − 1.09·53-s + 3·81-s + 2.11·89-s − 4.43·117-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.944596205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.944596205\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.789588214984616936995658760511, −8.580967856813785012333115446052, −8.218479137035831349038979197212, −7.87229092446838130368023295305, −7.34920351604880914188833212294, −7.01234917133769375689332390362, −6.45559539554391060175783871911, −6.12899400395768050817262831828, −5.80951898610700459269131207622, −5.61813737571389026957901027864, −5.12912111247385227923289009276, −4.67889714849769342898730033339, −3.98987411273445258550065834144, −3.68579418290606987770970817854, −3.31369348684660704816412464670, −2.94353723896191981835740042222, −2.29588520903813057044376099748, −1.86274877259394954740602944111, −1.08645371298573824041980669375, −0.48192076835468992320752258759,
0.48192076835468992320752258759, 1.08645371298573824041980669375, 1.86274877259394954740602944111, 2.29588520903813057044376099748, 2.94353723896191981835740042222, 3.31369348684660704816412464670, 3.68579418290606987770970817854, 3.98987411273445258550065834144, 4.67889714849769342898730033339, 5.12912111247385227923289009276, 5.61813737571389026957901027864, 5.80951898610700459269131207622, 6.12899400395768050817262831828, 6.45559539554391060175783871911, 7.01234917133769375689332390362, 7.34920351604880914188833212294, 7.87229092446838130368023295305, 8.218479137035831349038979197212, 8.580967856813785012333115446052, 8.789588214984616936995658760511
