| L(s) = 1 | − 1.46·3-s − 1.39·5-s + 2.18·7-s − 0.860·9-s + 11-s − 13-s + 2.04·15-s + 1.72·17-s − 6.58·19-s − 3.19·21-s − 6.78·23-s − 3.04·25-s + 5.64·27-s + 0.646·29-s − 10.3·31-s − 1.46·33-s − 3.05·35-s − 2.60·37-s + 1.46·39-s + 7.90·41-s − 11.5·43-s + 1.20·45-s + 4.79·47-s − 2.22·49-s − 2.51·51-s − 1.78·53-s − 1.39·55-s + ⋯ |
| L(s) = 1 | − 0.844·3-s − 0.625·5-s + 0.825·7-s − 0.286·9-s + 0.301·11-s − 0.277·13-s + 0.528·15-s + 0.417·17-s − 1.51·19-s − 0.697·21-s − 1.41·23-s − 0.609·25-s + 1.08·27-s + 0.120·29-s − 1.85·31-s − 0.254·33-s − 0.516·35-s − 0.427·37-s + 0.234·39-s + 1.23·41-s − 1.76·43-s + 0.179·45-s + 0.699·47-s − 0.318·49-s − 0.352·51-s − 0.245·53-s − 0.188·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 1.46T + 3T^{2} \) |
| 5 | \( 1 + 1.39T + 5T^{2} \) |
| 7 | \( 1 - 2.18T + 7T^{2} \) |
| 17 | \( 1 - 1.72T + 17T^{2} \) |
| 19 | \( 1 + 6.58T + 19T^{2} \) |
| 23 | \( 1 + 6.78T + 23T^{2} \) |
| 29 | \( 1 - 0.646T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 + 2.60T + 37T^{2} \) |
| 41 | \( 1 - 7.90T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 - 4.79T + 47T^{2} \) |
| 53 | \( 1 + 1.78T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 - 1.87T + 61T^{2} \) |
| 67 | \( 1 + 9.91T + 67T^{2} \) |
| 71 | \( 1 - 8.17T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 - 2.46T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 + 1.39T + 97T^{2} \) |
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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
−10.70076710947539021829709086759, −9.445410532376628199599892159410, −8.342572853793581189292111887096, −7.73204337375721425339821986733, −6.53356105343559798030098756010, −5.67089060545773837396201702477, −4.69044145089019047860163519554, −3.71743327183952501150014224131, −1.95186342124673537248040485256, 0,
1.95186342124673537248040485256, 3.71743327183952501150014224131, 4.69044145089019047860163519554, 5.67089060545773837396201702477, 6.53356105343559798030098756010, 7.73204337375721425339821986733, 8.342572853793581189292111887096, 9.445410532376628199599892159410, 10.70076710947539021829709086759
