| L(s) = 1 | + 3·5-s − 3·9-s − 3·11-s − 13-s − 12·17-s + 8·19-s + 3·23-s + 5·25-s − 3·29-s + 5·31-s + 4·37-s + 3·41-s + 43-s − 9·45-s − 9·47-s − 12·53-s − 9·55-s − 3·59-s − 13·61-s − 3·65-s + 7·67-s − 24·71-s + 20·73-s − 11·79-s + 9·81-s − 9·83-s − 36·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 9-s − 0.904·11-s − 0.277·13-s − 2.91·17-s + 1.83·19-s + 0.625·23-s + 25-s − 0.557·29-s + 0.898·31-s + 0.657·37-s + 0.468·41-s + 0.152·43-s − 1.34·45-s − 1.31·47-s − 1.64·53-s − 1.21·55-s − 0.390·59-s − 1.66·61-s − 0.372·65-s + 0.855·67-s − 2.84·71-s + 2.34·73-s − 1.23·79-s + 81-s − 0.987·83-s − 3.90·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.598968613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.598968613\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 11 T + 24 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−9.511250850547524667658744975281, −9.180235067807233258392473198288, −8.633322227471780737801032985010, −8.613516054753951119153771895427, −7.82966258576222510353527228687, −7.58810735598572486100210286387, −7.10023790597701166111981393015, −6.44533163813515332751528408770, −6.44467371891122426906962703035, −5.70652880416521864723764296952, −5.66442290022989852770416551410, −4.93140482931564978210486264923, −4.67524529839879655109708350273, −4.37857376437642729786415796441, −3.19645661435219207931314840825, −3.11194513590526314294780090565, −2.52625691503245215099659752697, −2.08540843382178549723940379934, −1.51480534531772980035154622036, −0.45163471260014307098992752618,
0.45163471260014307098992752618, 1.51480534531772980035154622036, 2.08540843382178549723940379934, 2.52625691503245215099659752697, 3.11194513590526314294780090565, 3.19645661435219207931314840825, 4.37857376437642729786415796441, 4.67524529839879655109708350273, 4.93140482931564978210486264923, 5.66442290022989852770416551410, 5.70652880416521864723764296952, 6.44467371891122426906962703035, 6.44533163813515332751528408770, 7.10023790597701166111981393015, 7.58810735598572486100210286387, 7.82966258576222510353527228687, 8.613516054753951119153771895427, 8.633322227471780737801032985010, 9.180235067807233258392473198288, 9.511250850547524667658744975281
