| L(s) = 1 | − 0.294·3-s + 5-s + 4.39·7-s − 2.91·9-s + 5.96·13-s − 0.294·15-s − 1.66·17-s − 7.69·19-s − 1.29·21-s + 0.904·23-s + 25-s + 1.74·27-s − 4.73·29-s + 5.12·31-s + 4.39·35-s − 0.184·37-s − 1.75·39-s − 2.62·41-s − 3.59·43-s − 2.91·45-s + 0.776·47-s + 12.2·49-s + 0.491·51-s − 9.59·53-s + 2.26·57-s + 11.2·59-s + 13.1·61-s + ⋯ |
| L(s) = 1 | − 0.170·3-s + 0.447·5-s + 1.65·7-s − 0.970·9-s + 1.65·13-s − 0.0761·15-s − 0.404·17-s − 1.76·19-s − 0.282·21-s + 0.188·23-s + 0.200·25-s + 0.335·27-s − 0.880·29-s + 0.920·31-s + 0.742·35-s − 0.0304·37-s − 0.281·39-s − 0.409·41-s − 0.548·43-s − 0.434·45-s + 0.113·47-s + 1.75·49-s + 0.0688·51-s − 1.31·53-s + 0.300·57-s + 1.46·59-s + 1.68·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.569837058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.569837058\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 0.294T + 3T^{2} \) |
| 7 | \( 1 - 4.39T + 7T^{2} \) |
| 13 | \( 1 - 5.96T + 13T^{2} \) |
| 17 | \( 1 + 1.66T + 17T^{2} \) |
| 19 | \( 1 + 7.69T + 19T^{2} \) |
| 23 | \( 1 - 0.904T + 23T^{2} \) |
| 29 | \( 1 + 4.73T + 29T^{2} \) |
| 31 | \( 1 - 5.12T + 31T^{2} \) |
| 37 | \( 1 + 0.184T + 37T^{2} \) |
| 41 | \( 1 + 2.62T + 41T^{2} \) |
| 43 | \( 1 + 3.59T + 43T^{2} \) |
| 47 | \( 1 - 0.776T + 47T^{2} \) |
| 53 | \( 1 + 9.59T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 7.79T + 67T^{2} \) |
| 71 | \( 1 - 6.97T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 + 4.09T + 83T^{2} \) |
| 89 | \( 1 + 0.466T + 89T^{2} \) |
| 97 | \( 1 - 7.09T + 97T^{2} \) |
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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
−8.016243344629260589032054054150, −6.77168565694678956965123973551, −6.34200580280418376245160345227, −5.56385047272457504790864829411, −5.04603280074808338643266217128, −4.24558210527535100893358209691, −3.53673678370780281546454787296, −2.33263668792997126303654540498, −1.81876649273418785517243507522, −0.78688036154394869688874867341,
0.78688036154394869688874867341, 1.81876649273418785517243507522, 2.33263668792997126303654540498, 3.53673678370780281546454787296, 4.24558210527535100893358209691, 5.04603280074808338643266217128, 5.56385047272457504790864829411, 6.34200580280418376245160345227, 6.77168565694678956965123973551, 8.016243344629260589032054054150
