| L(s) = 1 | + 3·5-s − 7-s + 11-s − 8·13-s + 4·17-s − 8·23-s + 5·25-s + 14·29-s + 11·31-s − 3·35-s − 4·37-s + 8·41-s + 4·43-s + 2·47-s − 6·49-s − 11·53-s + 3·55-s − 7·59-s − 10·61-s − 24·65-s + 10·67-s + 12·71-s + 6·73-s − 77-s + 11·79-s + 22·83-s + 12·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377·7-s + 0.301·11-s − 2.21·13-s + 0.970·17-s − 1.66·23-s + 25-s + 2.59·29-s + 1.97·31-s − 0.507·35-s − 0.657·37-s + 1.24·41-s + 0.609·43-s + 0.291·47-s − 6/7·49-s − 1.51·53-s + 0.404·55-s − 0.911·59-s − 1.28·61-s − 2.97·65-s + 1.22·67-s + 1.42·71-s + 0.702·73-s − 0.113·77-s + 1.23·79-s + 2.41·83-s + 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.754350339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.754350339\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 11 T + 68 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
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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
−9.356623024503091230549757367714, −9.355282157973774782968600113121, −8.505768901990037001125743037683, −7.972008114374747536753511027693, −7.971188354579998885771589021869, −7.49227201054561270605695988944, −6.78602654017249680443055734888, −6.51035163779180540364445163345, −6.23069127065640877659618388969, −5.95151603923943775622791506363, −5.20946396628294955268983296274, −4.92262947394355279709272895530, −4.68550916871114784992557492108, −4.10386684017211845325641681676, −3.35724879789769063992257528905, −2.92032600548400976930403380330, −2.33913554183567590112613766871, −2.22007343252318706293440562808, −1.30356431395739066919398836573, −0.61658825604699290437364686142,
0.61658825604699290437364686142, 1.30356431395739066919398836573, 2.22007343252318706293440562808, 2.33913554183567590112613766871, 2.92032600548400976930403380330, 3.35724879789769063992257528905, 4.10386684017211845325641681676, 4.68550916871114784992557492108, 4.92262947394355279709272895530, 5.20946396628294955268983296274, 5.95151603923943775622791506363, 6.23069127065640877659618388969, 6.51035163779180540364445163345, 6.78602654017249680443055734888, 7.49227201054561270605695988944, 7.971188354579998885771589021869, 7.972008114374747536753511027693, 8.505768901990037001125743037683, 9.355282157973774782968600113121, 9.356623024503091230549757367714
