| L(s) = 1 | − 9-s + 10·11-s − 4·16-s + 12·19-s − 4·29-s + 4·31-s + 18·41-s + 13·49-s + 10·61-s + 18·71-s + 6·79-s + 81-s − 22·89-s − 10·99-s − 8·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 3.01·11-s − 16-s + 2.75·19-s − 0.742·29-s + 0.718·31-s + 2.81·41-s + 13/7·49-s + 1.28·61-s + 2.13·71-s + 0.675·79-s + 1/9·81-s − 2.33·89-s − 1.00·99-s − 0.766·109-s + 4.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.0769·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.978131091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.978131091\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 81 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
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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
−9.996394008838985257822741622651, −9.633176476545315607029227176635, −9.243254900933984545463546409236, −9.217628409188435316700700330816, −8.740472855889520499932547426011, −8.187482288672853866644552945813, −7.42827363512632460370386271467, −7.39288382163713073087305494813, −6.84709493670711635863691803818, −6.25274741289702926900474348398, −6.18490496518153754585792650511, −5.32632479725822535632326487500, −5.16828518296412863150588415303, −4.25303219057324838174535968171, −3.86281532014586325850132885485, −3.73116461641501034639507186577, −2.79294761687251196328087093584, −2.32357140714888001353167288909, −1.16527492879914064662368930022, −1.06561585539677549750123855956,
1.06561585539677549750123855956, 1.16527492879914064662368930022, 2.32357140714888001353167288909, 2.79294761687251196328087093584, 3.73116461641501034639507186577, 3.86281532014586325850132885485, 4.25303219057324838174535968171, 5.16828518296412863150588415303, 5.32632479725822535632326487500, 6.18490496518153754585792650511, 6.25274741289702926900474348398, 6.84709493670711635863691803818, 7.39288382163713073087305494813, 7.42827363512632460370386271467, 8.187482288672853866644552945813, 8.740472855889520499932547426011, 9.217628409188435316700700330816, 9.243254900933984545463546409236, 9.633176476545315607029227176635, 9.996394008838985257822741622651
