| L(s) = 1 | + 2-s + 4-s − 2.87·5-s − 1.03·7-s + 8-s − 2.87·10-s − 4.17·11-s − 5.28·13-s − 1.03·14-s + 16-s + 2.95·17-s − 2.69·19-s − 2.87·20-s − 4.17·22-s − 0.151·23-s + 3.28·25-s − 5.28·26-s − 1.03·28-s + 2.97·29-s − 10.5·31-s + 32-s + 2.95·34-s + 2.97·35-s − 0.346·37-s − 2.69·38-s − 2.87·40-s + 5.88·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.28·5-s − 0.390·7-s + 0.353·8-s − 0.910·10-s − 1.25·11-s − 1.46·13-s − 0.276·14-s + 0.250·16-s + 0.715·17-s − 0.618·19-s − 0.643·20-s − 0.889·22-s − 0.0315·23-s + 0.657·25-s − 1.03·26-s − 0.195·28-s + 0.552·29-s − 1.90·31-s + 0.176·32-s + 0.506·34-s + 0.502·35-s − 0.0569·37-s − 0.437·38-s − 0.455·40-s + 0.918·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.247497385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.247497385\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 353 | \( 1 - T \) |
| good | 5 | \( 1 + 2.87T + 5T^{2} \) |
| 7 | \( 1 + 1.03T + 7T^{2} \) |
| 11 | \( 1 + 4.17T + 11T^{2} \) |
| 13 | \( 1 + 5.28T + 13T^{2} \) |
| 17 | \( 1 - 2.95T + 17T^{2} \) |
| 19 | \( 1 + 2.69T + 19T^{2} \) |
| 23 | \( 1 + 0.151T + 23T^{2} \) |
| 29 | \( 1 - 2.97T + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 + 0.346T + 37T^{2} \) |
| 41 | \( 1 - 5.88T + 41T^{2} \) |
| 43 | \( 1 - 3.70T + 43T^{2} \) |
| 47 | \( 1 - 3.93T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 - 2.07T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 6.67T + 67T^{2} \) |
| 71 | \( 1 + 1.87T + 71T^{2} \) |
| 73 | \( 1 - 7.79T + 73T^{2} \) |
| 79 | \( 1 - 5.83T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 - 0.0627T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−7.81398385175168028901103511397, −7.33790537392940311089575078328, −6.80319086755953860054286572954, −5.60320451272666637387308590843, −5.22202476790519732931416888527, −4.30758537098881663644002831355, −3.74283002590906223595912652744, −2.85595075657784936043712781715, −2.21656976859862385598223859011, −0.48654831571758657043081777606,
0.48654831571758657043081777606, 2.21656976859862385598223859011, 2.85595075657784936043712781715, 3.74283002590906223595912652744, 4.30758537098881663644002831355, 5.22202476790519732931416888527, 5.60320451272666637387308590843, 6.80319086755953860054286572954, 7.33790537392940311089575078328, 7.81398385175168028901103511397
