| L(s) = 1 | + 2-s − 2.69·3-s + 4-s + 2·5-s − 2.69·6-s − 0.693·7-s + 8-s + 4.25·9-s + 2·10-s + 11-s − 2.69·12-s − 4.25·13-s − 0.693·14-s − 5.38·15-s + 16-s + 7.95·17-s + 4.25·18-s + 2·20-s + 1.86·21-s + 22-s + 9.38·23-s − 2.69·24-s − 25-s − 4.25·26-s − 3.38·27-s − 0.693·28-s + 29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.55·3-s + 0.5·4-s + 0.894·5-s − 1.09·6-s − 0.262·7-s + 0.353·8-s + 1.41·9-s + 0.632·10-s + 0.301·11-s − 0.777·12-s − 1.18·13-s − 0.185·14-s − 1.39·15-s + 0.250·16-s + 1.92·17-s + 1.00·18-s + 0.447·20-s + 0.407·21-s + 0.213·22-s + 1.95·23-s − 0.549·24-s − 0.200·25-s − 0.834·26-s − 0.652·27-s − 0.131·28-s + 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.311755794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.311755794\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.69T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + 0.693T + 7T^{2} \) |
| 13 | \( 1 + 4.25T + 13T^{2} \) |
| 17 | \( 1 - 7.95T + 17T^{2} \) |
| 23 | \( 1 - 9.38T + 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 - 6.64T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 + 2.56T + 41T^{2} \) |
| 43 | \( 1 - 2.43T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 7.13T + 61T^{2} \) |
| 67 | \( 1 - 8.08T + 67T^{2} \) |
| 71 | \( 1 - 2.43T + 71T^{2} \) |
| 73 | \( 1 - 3.43T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + 3.82T + 83T^{2} \) |
| 89 | \( 1 - 8.90T + 89T^{2} \) |
| 97 | \( 1 + 6.90T + 97T^{2} \) |
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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
−7.40707562076016883789764812354, −6.85101162669711019841057013453, −6.36222735048332424203851301407, −5.53122408590765501699423491338, −5.21777414882370772995893214755, −4.73620179271286978403083221513, −3.55612164909932728432017826456, −2.78696612678683951857599673305, −1.63779376005893889972726705672, −0.76800017200529102891408070510,
0.76800017200529102891408070510, 1.63779376005893889972726705672, 2.78696612678683951857599673305, 3.55612164909932728432017826456, 4.73620179271286978403083221513, 5.21777414882370772995893214755, 5.53122408590765501699423491338, 6.36222735048332424203851301407, 6.85101162669711019841057013453, 7.40707562076016883789764812354
