L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 2·11-s + 4·13-s + 15-s + 2·17-s − 2·19-s − 21-s + 4·23-s + 25-s + 27-s − 2·29-s − 6·31-s + 2·33-s − 35-s − 6·37-s + 4·39-s + 6·41-s − 4·43-s + 45-s + 49-s + 2·51-s + 8·53-s + 2·55-s − 2·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 1.10·13-s + 0.258·15-s + 0.485·17-s − 0.458·19-s − 0.218·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.07·31-s + 0.348·33-s − 0.169·35-s − 0.986·37-s + 0.640·39-s + 0.937·41-s − 0.609·43-s + 0.149·45-s + 1/7·49-s + 0.280·51-s + 1.09·53-s + 0.269·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.830744333\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.830744333\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−11.05601917102722003252352909039, −10.26190288946588984991947102908, −9.181203462215863258681403979351, −8.735538816120639668028034573127, −7.48284568639593696898416392075, −6.51502067396389468143134947504, −5.55375053090780051552954501584, −4.08706078179886277200739673348, −3.08927995906889680200761478861, −1.54792096213768469866947757162,
1.54792096213768469866947757162, 3.08927995906889680200761478861, 4.08706078179886277200739673348, 5.55375053090780051552954501584, 6.51502067396389468143134947504, 7.48284568639593696898416392075, 8.735538816120639668028034573127, 9.181203462215863258681403979351, 10.26190288946588984991947102908, 11.05601917102722003252352909039