| L(s) = 1 | − 2.58·2-s + 4.66·4-s + 4.25·5-s − 1.55·7-s − 6.87·8-s − 10.9·10-s + 1.50·11-s − 0.166·13-s + 4.00·14-s + 8.41·16-s + 7.68·17-s + 7.82·19-s + 19.8·20-s − 3.88·22-s − 1.82·23-s + 13.0·25-s + 0.430·26-s − 7.24·28-s − 2.12·29-s − 5.38·31-s − 7.96·32-s − 19.8·34-s − 6.60·35-s + 3.94·37-s − 20.1·38-s − 29.2·40-s + 0.332·41-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 2.33·4-s + 1.90·5-s − 0.587·7-s − 2.42·8-s − 3.47·10-s + 0.453·11-s − 0.0462·13-s + 1.07·14-s + 2.10·16-s + 1.86·17-s + 1.79·19-s + 4.43·20-s − 0.828·22-s − 0.379·23-s + 2.61·25-s + 0.0844·26-s − 1.36·28-s − 0.394·29-s − 0.967·31-s − 1.40·32-s − 3.40·34-s − 1.11·35-s + 0.649·37-s − 3.27·38-s − 4.62·40-s + 0.0519·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3447 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3447 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.315571718\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.315571718\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 2.58T + 2T^{2} \) |
| 5 | \( 1 - 4.25T + 5T^{2} \) |
| 7 | \( 1 + 1.55T + 7T^{2} \) |
| 11 | \( 1 - 1.50T + 11T^{2} \) |
| 13 | \( 1 + 0.166T + 13T^{2} \) |
| 17 | \( 1 - 7.68T + 17T^{2} \) |
| 19 | \( 1 - 7.82T + 19T^{2} \) |
| 23 | \( 1 + 1.82T + 23T^{2} \) |
| 29 | \( 1 + 2.12T + 29T^{2} \) |
| 31 | \( 1 + 5.38T + 31T^{2} \) |
| 37 | \( 1 - 3.94T + 37T^{2} \) |
| 41 | \( 1 - 0.332T + 41T^{2} \) |
| 43 | \( 1 - 4.68T + 43T^{2} \) |
| 47 | \( 1 - 1.24T + 47T^{2} \) |
| 53 | \( 1 - 9.61T + 53T^{2} \) |
| 59 | \( 1 + 15.2T + 59T^{2} \) |
| 61 | \( 1 + 4.86T + 61T^{2} \) |
| 67 | \( 1 - 9.31T + 67T^{2} \) |
| 71 | \( 1 + 7.05T + 71T^{2} \) |
| 73 | \( 1 - 8.88T + 73T^{2} \) |
| 79 | \( 1 + 0.129T + 79T^{2} \) |
| 83 | \( 1 + 3.08T + 83T^{2} \) |
| 89 | \( 1 - 9.24T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−9.008188112645419868563102763791, −7.85131138133160939842025822881, −7.33662713039683649296059892198, −6.47522147284990561090400018073, −5.84809507227892501938603750626, −5.31137130163820863583438529770, −3.38572057828928299612661245061, −2.62619806927675802878845394206, −1.60463807700442439322729927419, −0.983905074893334390392783251453,
0.983905074893334390392783251453, 1.60463807700442439322729927419, 2.62619806927675802878845394206, 3.38572057828928299612661245061, 5.31137130163820863583438529770, 5.84809507227892501938603750626, 6.47522147284990561090400018073, 7.33662713039683649296059892198, 7.85131138133160939842025822881, 9.008188112645419868563102763791
