| L(s) = 1 | + 3-s + 0.618·7-s − 2·9-s + 3·11-s + 3.47·13-s − 7.85·17-s + 2.76·19-s + 0.618·21-s − 4.85·23-s − 5·27-s − 6.70·29-s + 10.2·31-s + 3·33-s − 5.09·37-s + 3.47·39-s + 3.70·41-s − 0.909·43-s − 3·47-s − 6.61·49-s − 7.85·51-s + 0.708·53-s + 2.76·57-s − 6.70·59-s − 10.2·61-s − 1.23·63-s + 11.4·67-s − 4.85·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.233·7-s − 0.666·9-s + 0.904·11-s + 0.962·13-s − 1.90·17-s + 0.634·19-s + 0.134·21-s − 1.01·23-s − 0.962·27-s − 1.24·29-s + 1.83·31-s + 0.522·33-s − 0.836·37-s + 0.555·39-s + 0.579·41-s − 0.138·43-s − 0.437·47-s − 0.945·49-s − 1.09·51-s + 0.0972·53-s + 0.366·57-s − 0.873·59-s − 1.31·61-s − 0.155·63-s + 1.40·67-s − 0.584·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 - 0.618T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 + 7.85T + 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 23 | \( 1 + 4.85T + 23T^{2} \) |
| 29 | \( 1 + 6.70T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 5.09T + 37T^{2} \) |
| 41 | \( 1 - 3.70T + 41T^{2} \) |
| 43 | \( 1 + 0.909T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 - 0.708T + 53T^{2} \) |
| 59 | \( 1 + 6.70T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 - 1.23T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 - 1.85T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−7.40768798810880869788666946766, −6.40639459512993552468807912570, −6.24572668513549829699939532842, −5.24948403442205478850532309724, −4.36827106337921501663495044598, −3.80647894834641104181970888358, −3.02994019096276361634607795766, −2.15985516043586251088003513496, −1.39680895153490654288642854389, 0,
1.39680895153490654288642854389, 2.15985516043586251088003513496, 3.02994019096276361634607795766, 3.80647894834641104181970888358, 4.36827106337921501663495044598, 5.24948403442205478850532309724, 6.24572668513549829699939532842, 6.40639459512993552468807912570, 7.40768798810880869788666946766
