| L(s) = 1 | + 4·3-s + 5·5-s + 27·9-s + 60·11-s − 172·13-s + 20·15-s + 18·17-s + 44·19-s − 48·23-s + 260·27-s − 372·29-s + 176·31-s + 240·33-s − 254·37-s − 688·39-s − 372·41-s − 200·43-s + 135·45-s + 168·47-s + 72·51-s + 498·53-s + 300·55-s + 176·57-s − 252·59-s − 58·61-s − 860·65-s + 1.03e3·67-s + ⋯ |
| L(s) = 1 | + 0.769·3-s + 0.447·5-s + 9-s + 1.64·11-s − 3.66·13-s + 0.344·15-s + 0.256·17-s + 0.531·19-s − 0.435·23-s + 1.85·27-s − 2.38·29-s + 1.01·31-s + 1.26·33-s − 1.12·37-s − 2.82·39-s − 1.41·41-s − 0.709·43-s + 0.447·45-s + 0.521·47-s + 0.197·51-s + 1.29·53-s + 0.735·55-s + 0.408·57-s − 0.556·59-s − 0.121·61-s − 1.64·65-s + 1.88·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.399242859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.399242859\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T - 11 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 60 T + 2269 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 86 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T - 4589 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 44 T - 4923 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 48 T - 9863 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 186 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 176 T + 1185 T^{2} - 176 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 254 T + 13863 T^{2} + 254 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 186 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 100 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 168 T - 75599 T^{2} - 168 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 498 T + 99127 T^{2} - 498 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 252 T - 141875 T^{2} + 252 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 58 T - 223617 T^{2} + 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 1036 T + 772533 T^{2} - 1036 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 168 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 506 T - 132981 T^{2} - 506 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 272 T - 419055 T^{2} + 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 948 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 1014 T + 323227 T^{2} + 1014 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 766 T + p^{3} T^{2} )^{2} \) |
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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
−9.729286940912639186288375549760, −9.414837248999068069032615846532, −9.325244976210079359593563721425, −8.456505242091584106074258470348, −8.341565446525374288992805923732, −7.57429498570608488850016446446, −7.24396243222336478691446717079, −6.91910388234299659935407845349, −6.80051036420281735703027919894, −5.96562870077476561767770474943, −5.30465730930715664423313929007, −5.01593860549979173387449151354, −4.63267935103522204296159263675, −3.81809269215653066383738330233, −3.73499535039986700903981610021, −2.67583488708000378409213235899, −2.56108399182243948839960056833, −1.77747387553424690364463207866, −1.40947913267018933548670598600, −0.35136011162668215156990800766,
0.35136011162668215156990800766, 1.40947913267018933548670598600, 1.77747387553424690364463207866, 2.56108399182243948839960056833, 2.67583488708000378409213235899, 3.73499535039986700903981610021, 3.81809269215653066383738330233, 4.63267935103522204296159263675, 5.01593860549979173387449151354, 5.30465730930715664423313929007, 5.96562870077476561767770474943, 6.80051036420281735703027919894, 6.91910388234299659935407845349, 7.24396243222336478691446717079, 7.57429498570608488850016446446, 8.341565446525374288992805923732, 8.456505242091584106074258470348, 9.325244976210079359593563721425, 9.414837248999068069032615846532, 9.729286940912639186288375549760
