| L(s) = 1 | − 9-s + 8·11-s − 8·19-s + 4·29-s + 16·31-s + 4·41-s − 49-s + 24·59-s + 12·61-s + 16·79-s + 81-s − 4·89-s − 8·99-s − 20·101-s + 4·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 8·171-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 2.41·11-s − 1.83·19-s + 0.742·29-s + 2.87·31-s + 0.624·41-s − 1/7·49-s + 3.12·59-s + 1.53·61-s + 1.80·79-s + 1/9·81-s − 0.423·89-s − 0.804·99-s − 1.99·101-s + 0.383·109-s + 2.36·121-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 + 0.611·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.003743019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.003743019\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.478217324568364000488532062647, −8.454969657089623230950051376659, −7.978955217923789325811221603688, −7.53866662088482774447940518004, −6.75338635173636874775582034647, −6.73837378818321971150503245788, −6.40819951698755898565464213633, −6.37785003830148925790951108271, −5.53657259099073031288880385471, −5.41351214295819207011464321839, −4.67405637386207654270044733715, −4.34050874072455264043169224503, −3.95677214671702978043301025888, −3.92850059292196744680162442417, −3.08136921760493298379756611318, −2.74503096587142595198191394088, −2.14321323203838435653979870183, −1.77842839934051188583083472097, −0.845080311162930781375401201501, −0.802153116721175301060501255618,
0.802153116721175301060501255618, 0.845080311162930781375401201501, 1.77842839934051188583083472097, 2.14321323203838435653979870183, 2.74503096587142595198191394088, 3.08136921760493298379756611318, 3.92850059292196744680162442417, 3.95677214671702978043301025888, 4.34050874072455264043169224503, 4.67405637386207654270044733715, 5.41351214295819207011464321839, 5.53657259099073031288880385471, 6.37785003830148925790951108271, 6.40819951698755898565464213633, 6.73837378818321971150503245788, 6.75338635173636874775582034647, 7.53866662088482774447940518004, 7.978955217923789325811221603688, 8.454969657089623230950051376659, 8.478217324568364000488532062647
