| L(s) = 1 | + 1.14·3-s − 1.34·5-s + 7-s − 1.68·9-s − 1.14·11-s + 13-s − 1.53·15-s + 5.83·17-s + 3.34·19-s + 1.14·21-s + 3.17·23-s − 3.19·25-s − 5.37·27-s + 10.4·29-s − 1.63·31-s − 1.31·33-s − 1.34·35-s + 8.51·37-s + 1.14·39-s − 0.292·41-s + 8.15·43-s + 2.26·45-s + 10.6·47-s + 49-s + 6.68·51-s − 0.782·53-s + 1.53·55-s + ⋯ |
| L(s) = 1 | + 0.661·3-s − 0.600·5-s + 0.377·7-s − 0.561·9-s − 0.345·11-s + 0.277·13-s − 0.397·15-s + 1.41·17-s + 0.766·19-s + 0.250·21-s + 0.662·23-s − 0.639·25-s − 1.03·27-s + 1.94·29-s − 0.293·31-s − 0.228·33-s − 0.226·35-s + 1.40·37-s + 0.183·39-s − 0.0457·41-s + 1.24·43-s + 0.337·45-s + 1.54·47-s + 0.142·49-s + 0.936·51-s − 0.107·53-s + 0.207·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.963304779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.963304779\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.14T + 3T^{2} \) |
| 5 | \( 1 + 1.34T + 5T^{2} \) |
| 11 | \( 1 + 1.14T + 11T^{2} \) |
| 17 | \( 1 - 5.83T + 17T^{2} \) |
| 19 | \( 1 - 3.34T + 19T^{2} \) |
| 23 | \( 1 - 3.17T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 + 1.63T + 31T^{2} \) |
| 37 | \( 1 - 8.51T + 37T^{2} \) |
| 41 | \( 1 + 0.292T + 41T^{2} \) |
| 43 | \( 1 - 8.15T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 + 0.782T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 6.10T + 67T^{2} \) |
| 71 | \( 1 + 1.53T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 + 0.882T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 5.73T + 89T^{2} \) |
| 97 | \( 1 + 5.34T + 97T^{2} \) |
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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
−9.361708696516279226457851260679, −8.660910726821436587712448421628, −7.70796352454159055241506083562, −7.64357321295259739145065555352, −6.16142811753794880321520619207, −5.37184436930791769963718952331, −4.33017221439190916384393138547, −3.31836996127260909596993916551, −2.61236956096011122294262688049, −1.01498067963245274444028235850,
1.01498067963245274444028235850, 2.61236956096011122294262688049, 3.31836996127260909596993916551, 4.33017221439190916384393138547, 5.37184436930791769963718952331, 6.16142811753794880321520619207, 7.64357321295259739145065555352, 7.70796352454159055241506083562, 8.660910726821436587712448421628, 9.361708696516279226457851260679
