| L(s) = 1 | + 0.440·3-s + 0.359·5-s + 0.921·7-s − 2.80·9-s + 1.53·11-s − 2.01·13-s + 0.158·15-s − 1.43·17-s + 19-s + 0.406·21-s + 4.72·23-s − 4.87·25-s − 2.55·27-s − 7.88·29-s + 1.20·31-s + 0.674·33-s + 0.331·35-s − 8.80·37-s − 0.887·39-s − 4.83·41-s + 5.59·43-s − 1.00·45-s + 12.1·47-s − 6.14·49-s − 0.631·51-s − 53-s + 0.549·55-s + ⋯ |
| L(s) = 1 | + 0.254·3-s + 0.160·5-s + 0.348·7-s − 0.935·9-s + 0.461·11-s − 0.558·13-s + 0.0408·15-s − 0.347·17-s + 0.229·19-s + 0.0886·21-s + 0.984·23-s − 0.974·25-s − 0.492·27-s − 1.46·29-s + 0.215·31-s + 0.117·33-s + 0.0560·35-s − 1.44·37-s − 0.142·39-s − 0.755·41-s + 0.852·43-s − 0.150·45-s + 1.77·47-s − 0.878·49-s − 0.0883·51-s − 0.137·53-s + 0.0741·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| good | 3 | \( 1 - 0.440T + 3T^{2} \) |
| 5 | \( 1 - 0.359T + 5T^{2} \) |
| 7 | \( 1 - 0.921T + 7T^{2} \) |
| 11 | \( 1 - 1.53T + 11T^{2} \) |
| 13 | \( 1 + 2.01T + 13T^{2} \) |
| 17 | \( 1 + 1.43T + 17T^{2} \) |
| 23 | \( 1 - 4.72T + 23T^{2} \) |
| 29 | \( 1 + 7.88T + 29T^{2} \) |
| 31 | \( 1 - 1.20T + 31T^{2} \) |
| 37 | \( 1 + 8.80T + 37T^{2} \) |
| 41 | \( 1 + 4.83T + 41T^{2} \) |
| 43 | \( 1 - 5.59T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 + 0.555T + 61T^{2} \) |
| 67 | \( 1 - 5.72T + 67T^{2} \) |
| 71 | \( 1 + 8.58T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 4.92T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 0.927T + 97T^{2} \) |
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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
−8.059446277538393030522449514173, −7.42366347674719029374951725622, −6.66552483787309994168606296987, −5.72166286541780292647513364012, −5.21558787381415119931069514415, −4.21486030743091153590507810079, −3.35976919582148112621375734867, −2.47298434736153737709741649804, −1.55372424212605988225802541898, 0,
1.55372424212605988225802541898, 2.47298434736153737709741649804, 3.35976919582148112621375734867, 4.21486030743091153590507810079, 5.21558787381415119931069514415, 5.72166286541780292647513364012, 6.66552483787309994168606296987, 7.42366347674719029374951725622, 8.059446277538393030522449514173
