| L(s) = 1 | − 4·11-s − 4·16-s − 2·19-s − 4·29-s − 20·41-s + 5·49-s + 16·59-s + 14·61-s + 4·71-s + 6·79-s + 24·89-s + 20·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 16·176-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1.20·11-s − 16-s − 0.458·19-s − 0.742·29-s − 3.12·41-s + 5/7·49-s + 2.08·59-s + 1.79·61-s + 0.474·71-s + 0.675·79-s + 2.54·89-s + 1.91·109-s − 0.909·121-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 + 1/13·169-s + 0.0760·173-s + 1.20·176-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.160338811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.160338811\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 49 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 121 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
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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
−10.87240434242808781512464123212, −10.14829794198471101193469294594, −9.991052788747482762643905688593, −9.527989392188956045923390968255, −8.815020286907300148411750285062, −8.578465753594531821064127410577, −8.245454353155008448589258464159, −7.60980095317721333550214184179, −7.23026675953908996170566052590, −6.69727444983279000418164253367, −6.43670181984921703174629609942, −5.61607273181293845511805862691, −5.24311646809836481901098686769, −4.90785064486306651628642088158, −4.22519080769197338080037363099, −3.62851974131635296226806583827, −3.11063340517579017408229179528, −2.18245715759109332964627049509, −2.02319599166729767438112215176, −0.55234322365171269888385838808,
0.55234322365171269888385838808, 2.02319599166729767438112215176, 2.18245715759109332964627049509, 3.11063340517579017408229179528, 3.62851974131635296226806583827, 4.22519080769197338080037363099, 4.90785064486306651628642088158, 5.24311646809836481901098686769, 5.61607273181293845511805862691, 6.43670181984921703174629609942, 6.69727444983279000418164253367, 7.23026675953908996170566052590, 7.60980095317721333550214184179, 8.245454353155008448589258464159, 8.578465753594531821064127410577, 8.815020286907300148411750285062, 9.527989392188956045923390968255, 9.991052788747482762643905688593, 10.14829794198471101193469294594, 10.87240434242808781512464123212
