| L(s) = 1 | + 2.21·3-s − 5-s − 7-s + 1.90·9-s − 11-s + 0.836·13-s − 2.21·15-s − 2.83·17-s + 2·19-s − 2.21·21-s − 1.52·23-s + 25-s − 2.42·27-s − 0.755·29-s + 5.31·31-s − 2.21·33-s + 35-s − 5.33·37-s + 1.85·39-s + 0.688·41-s − 10.5·43-s − 1.90·45-s − 5.88·47-s + 49-s − 6.28·51-s + 4.08·53-s + 55-s + ⋯ |
| L(s) = 1 | + 1.27·3-s − 0.447·5-s − 0.377·7-s + 0.634·9-s − 0.301·11-s + 0.232·13-s − 0.571·15-s − 0.687·17-s + 0.458·19-s − 0.483·21-s − 0.318·23-s + 0.200·25-s − 0.467·27-s − 0.140·29-s + 0.953·31-s − 0.385·33-s + 0.169·35-s − 0.876·37-s + 0.296·39-s + 0.107·41-s − 1.60·43-s − 0.283·45-s − 0.858·47-s + 0.142·49-s − 0.879·51-s + 0.561·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2.21T + 3T^{2} \) |
| 13 | \( 1 - 0.836T + 13T^{2} \) |
| 17 | \( 1 + 2.83T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 1.52T + 23T^{2} \) |
| 29 | \( 1 + 0.755T + 29T^{2} \) |
| 31 | \( 1 - 5.31T + 31T^{2} \) |
| 37 | \( 1 + 5.33T + 37T^{2} \) |
| 41 | \( 1 - 0.688T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 5.88T + 47T^{2} \) |
| 53 | \( 1 - 4.08T + 53T^{2} \) |
| 59 | \( 1 + 5.87T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 + 6.90T + 67T^{2} \) |
| 71 | \( 1 - 2.75T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 2.42T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 + 18.2T + 97T^{2} \) |
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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
−7.86549803875960077357191044606, −7.13068565141456223513129855378, −6.48894118425910511356522531543, −5.52942669860822318439828139821, −4.62897754463193798299796659502, −3.82025575332545048729545997489, −3.18133508177250812198957866065, −2.52310841909851615046170301107, −1.53372303917013645193145559515, 0,
1.53372303917013645193145559515, 2.52310841909851615046170301107, 3.18133508177250812198957866065, 3.82025575332545048729545997489, 4.62897754463193798299796659502, 5.52942669860822318439828139821, 6.48894118425910511356522531543, 7.13068565141456223513129855378, 7.86549803875960077357191044606
