L(s) = 1 | − 2-s + 4-s + 3·5-s + 7-s − 8-s − 3·10-s + 11-s − 13-s − 14-s + 16-s + 5·17-s − 19-s + 3·20-s − 22-s − 23-s + 4·25-s + 26-s + 28-s + 7·29-s − 2·31-s − 32-s − 5·34-s + 3·35-s + 37-s + 38-s − 3·40-s + 6·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.377·7-s − 0.353·8-s − 0.948·10-s + 0.301·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 1.21·17-s − 0.229·19-s + 0.670·20-s − 0.213·22-s − 0.208·23-s + 4/5·25-s + 0.196·26-s + 0.188·28-s + 1.29·29-s − 0.359·31-s − 0.176·32-s − 0.857·34-s + 0.507·35-s + 0.164·37-s + 0.162·38-s − 0.474·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.761104613\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.761104613\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.410330285667180458900953642361, −8.747177757509462895290865415839, −7.86550414420019729808395348479, −7.06048295349616083721798972803, −6.09215447957051242590274646223, −5.58029949714007967425702896846, −4.47695299580006008543989778833, −3.05971025238251891310798552403, −2.07649822353842967948528730142, −1.11293587495685660027839062390,
1.11293587495685660027839062390, 2.07649822353842967948528730142, 3.05971025238251891310798552403, 4.47695299580006008543989778833, 5.58029949714007967425702896846, 6.09215447957051242590274646223, 7.06048295349616083721798972803, 7.86550414420019729808395348479, 8.747177757509462895290865415839, 9.410330285667180458900953642361