L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s + 9-s − 2·14-s + 16-s + 4·17-s + 18-s + 16·23-s − 6·25-s − 2·28-s + 32-s + 4·34-s + 36-s − 12·41-s + 16·46-s + 3·49-s − 6·50-s − 2·56-s − 2·63-s + 64-s + 4·68-s + 16·71-s + 72-s + 20·73-s + 81-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.534·14-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 3.33·23-s − 6/5·25-s − 0.377·28-s + 0.176·32-s + 0.685·34-s + 1/6·36-s − 1.87·41-s + 2.35·46-s + 3/7·49-s − 0.848·50-s − 0.267·56-s − 0.251·63-s + 1/8·64-s + 0.485·68-s + 1.89·71-s + 0.117·72-s + 2.34·73-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56448 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56448 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.135238861\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.135238861\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−10.03376970334634580161769057016, −9.458622000763330506703484219944, −9.217470378406294805238498141833, −8.356918966632091922574869118017, −7.941030481552254497163330431349, −7.14585602865319249961807679325, −6.84552480602986516155667793601, −6.43557747993001049134741297977, −5.40100953109829334895217907278, −5.33985014787602837985094112170, −4.54031360247962128301261767712, −3.62482887081886485478101246558, −3.29683272363669349411451437037, −2.47888758091438588042078740356, −1.23111713389498587746226154587,
1.23111713389498587746226154587, 2.47888758091438588042078740356, 3.29683272363669349411451437037, 3.62482887081886485478101246558, 4.54031360247962128301261767712, 5.33985014787602837985094112170, 5.40100953109829334895217907278, 6.43557747993001049134741297977, 6.84552480602986516155667793601, 7.14585602865319249961807679325, 7.941030481552254497163330431349, 8.356918966632091922574869118017, 9.217470378406294805238498141833, 9.458622000763330506703484219944, 10.03376970334634580161769057016