| L(s) = 1 | + 4·5-s + 4·7-s − 3·9-s + 8·11-s − 3·13-s − 7·17-s − 5·19-s + 8·23-s + 2·25-s + 29-s + 3·31-s + 16·35-s − 11·37-s + 9·41-s − 5·43-s − 12·45-s − 3·47-s + 9·49-s − 3·53-s + 32·55-s + 7·59-s − 3·61-s − 12·63-s − 12·65-s − 13·67-s − 16·71-s − 7·73-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1.51·7-s − 9-s + 2.41·11-s − 0.832·13-s − 1.69·17-s − 1.14·19-s + 1.66·23-s + 2/5·25-s + 0.185·29-s + 0.538·31-s + 2.70·35-s − 1.80·37-s + 1.40·41-s − 0.762·43-s − 1.78·45-s − 0.437·47-s + 9/7·49-s − 0.412·53-s + 4.31·55-s + 0.911·59-s − 0.384·61-s − 1.51·63-s − 1.48·65-s − 1.58·67-s − 1.89·71-s − 0.819·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.212449265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.212449265\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 3 T - 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 9 T + 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + T - 82 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 17 T + 192 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
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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
−11.95764646429721083201520755685, −11.86587546310485270159398137294, −11.33106801869660453158138901078, −10.92292701491041295967004508239, −10.42828064234449101434286713912, −9.860824028105187945095381955488, −9.140408818947289994260450894603, −8.928448464120350357956573327813, −8.781905920340361212785597792389, −8.026776875535515146757411981750, −7.06579334550312997093215539801, −6.81126289867616638219070776844, −6.01452666127484331915187606485, −5.96273255125290601906640269212, −4.88020989592490665637607255954, −4.71891051159073265784383593327, −3.91098721878599581095103837476, −2.75271822423038963260300335811, −1.97229805884299553147153726157, −1.54870080446495394112970632088,
1.54870080446495394112970632088, 1.97229805884299553147153726157, 2.75271822423038963260300335811, 3.91098721878599581095103837476, 4.71891051159073265784383593327, 4.88020989592490665637607255954, 5.96273255125290601906640269212, 6.01452666127484331915187606485, 6.81126289867616638219070776844, 7.06579334550312997093215539801, 8.026776875535515146757411981750, 8.781905920340361212785597792389, 8.928448464120350357956573327813, 9.140408818947289994260450894603, 9.860824028105187945095381955488, 10.42828064234449101434286713912, 10.92292701491041295967004508239, 11.33106801869660453158138901078, 11.86587546310485270159398137294, 11.95764646429721083201520755685
