L(s) = 1 | + 3-s + 2·5-s − 7-s + 9-s + 4·11-s − 6·13-s + 2·15-s + 2·17-s + 4·19-s − 21-s + 8·23-s − 25-s + 27-s + 2·29-s + 4·33-s − 2·35-s + 10·37-s − 6·39-s − 6·41-s + 4·43-s + 2·45-s + 49-s + 2·51-s − 6·53-s + 8·55-s + 4·57-s − 4·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 1.66·13-s + 0.516·15-s + 0.485·17-s + 0.917·19-s − 0.218·21-s + 1.66·23-s − 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.696·33-s − 0.338·35-s + 1.64·37-s − 0.960·39-s − 0.937·41-s + 0.609·43-s + 0.298·45-s + 1/7·49-s + 0.280·51-s − 0.824·53-s + 1.07·55-s + 0.529·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.457512464\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.457512464\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.458622000763330506703484219944, −9.217470378406294805238498141833, −7.941030481552254497163330431349, −7.14585602865319249961807679325, −6.43557747993001049134741297977, −5.40100953109829334895217907278, −4.54031360247962128301261767712, −3.29683272363669349411451437037, −2.47888758091438588042078740356, −1.23111713389498587746226154587,
1.23111713389498587746226154587, 2.47888758091438588042078740356, 3.29683272363669349411451437037, 4.54031360247962128301261767712, 5.40100953109829334895217907278, 6.43557747993001049134741297977, 7.14585602865319249961807679325, 7.941030481552254497163330431349, 9.217470378406294805238498141833, 9.458622000763330506703484219944