| L(s) = 1 | − 3-s + 5-s + 9-s − 4·11-s + 13-s − 15-s + 17-s − 8·19-s + 25-s − 27-s + 6·29-s − 8·31-s + 4·33-s − 2·37-s − 39-s − 10·41-s − 4·43-s + 45-s − 7·49-s − 51-s + 6·53-s − 4·55-s + 8·57-s + 6·61-s + 65-s − 4·67-s − 8·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 0.258·15-s + 0.242·17-s − 1.83·19-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.696·33-s − 0.328·37-s − 0.160·39-s − 1.56·41-s − 0.609·43-s + 0.149·45-s − 49-s − 0.140·51-s + 0.824·53-s − 0.539·55-s + 1.05·57-s + 0.768·61-s + 0.124·65-s − 0.488·67-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.24256342343716, −13.08204567529739, −12.62151027472653, −12.08185615081175, −11.62776243184731, −11.03604366704561, −10.55254864092074, −10.34678188321170, −9.949036068875706, −9.241441657830651, −8.673694926305373, −8.312746526841131, −7.874667955345021, −7.019129566239680, −6.875140016204470, −6.166516591446215, −5.804028431280298, −5.178380346464678, −4.896248759676506, −4.224814046528993, −3.660871944552777, −2.951680930854178, −2.401575374576535, −1.774959306270927, −1.239952181072895, 0, 0,
1.239952181072895, 1.774959306270927, 2.401575374576535, 2.951680930854178, 3.660871944552777, 4.224814046528993, 4.896248759676506, 5.178380346464678, 5.804028431280298, 6.166516591446215, 6.875140016204470, 7.019129566239680, 7.874667955345021, 8.312746526841131, 8.673694926305373, 9.241441657830651, 9.949036068875706, 10.34678188321170, 10.55254864092074, 11.03604366704561, 11.62776243184731, 12.08185615081175, 12.62151027472653, 13.08204567529739, 13.24256342343716
