| L(s) = 1 | − 2.24·3-s + 0.529·5-s + 7-s + 2.05·9-s + 2.24·11-s + 13-s − 1.19·15-s − 1.30·17-s + 1.47·19-s − 2.24·21-s − 5.83·23-s − 4.71·25-s + 2.11·27-s + 5.22·29-s + 7.02·31-s − 5.05·33-s + 0.529·35-s − 2.36·37-s − 2.24·39-s + 6.49·41-s − 11.3·43-s + 1.08·45-s − 8.58·47-s + 49-s + 2.94·51-s + 11.2·53-s + 1.19·55-s + ⋯ |
| L(s) = 1 | − 1.29·3-s + 0.236·5-s + 0.377·7-s + 0.686·9-s + 0.678·11-s + 0.277·13-s − 0.307·15-s − 0.317·17-s + 0.337·19-s − 0.490·21-s − 1.21·23-s − 0.943·25-s + 0.407·27-s + 0.969·29-s + 1.26·31-s − 0.880·33-s + 0.0894·35-s − 0.389·37-s − 0.360·39-s + 1.01·41-s − 1.73·43-s + 0.162·45-s − 1.25·47-s + 0.142·49-s + 0.411·51-s + 1.54·53-s + 0.160·55-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
| \(L(1)\) |
\(\approx\) |
\(1.118889316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.118889316\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.24T + 3T^{2} \) |
| 5 | \( 1 - 0.529T + 5T^{2} \) |
| 11 | \( 1 - 2.24T + 11T^{2} \) |
| 17 | \( 1 + 1.30T + 17T^{2} \) |
| 19 | \( 1 - 1.47T + 19T^{2} \) |
| 23 | \( 1 + 5.83T + 23T^{2} \) |
| 29 | \( 1 - 5.22T + 29T^{2} \) |
| 31 | \( 1 - 7.02T + 31T^{2} \) |
| 37 | \( 1 + 2.36T + 37T^{2} \) |
| 41 | \( 1 - 6.49T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 8.58T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 15.9T + 67T^{2} \) |
| 71 | \( 1 + 1.19T + 71T^{2} \) |
| 73 | \( 1 - 7.64T + 73T^{2} \) |
| 79 | \( 1 - 1.33T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 - 6.91T + 89T^{2} \) |
| 97 | \( 1 + 3.47T + 97T^{2} \) |
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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
−9.818397072565604628320767834959, −8.650363278072362192834621545644, −7.946585398528833153721002390058, −6.70802981775855558616475621121, −6.29115308760223016850462173196, −5.42230861252020864753043610279, −4.65725148729726614126099270986, −3.68059238639301667916185921862, −2.11654988540624096334527919707, −0.822900904814127900151644335214,
0.822900904814127900151644335214, 2.11654988540624096334527919707, 3.68059238639301667916185921862, 4.65725148729726614126099270986, 5.42230861252020864753043610279, 6.29115308760223016850462173196, 6.70802981775855558616475621121, 7.946585398528833153721002390058, 8.650363278072362192834621545644, 9.818397072565604628320767834959
