| L(s) = 1 | + 2.85·7-s − 1.85·11-s − 0.854·13-s + 1.14·17-s + 2·19-s − 4.85·23-s − 3.70·29-s + 2.70·31-s + 5.85·37-s + 11.5·41-s − 0.854·43-s − 6.70·47-s + 1.14·49-s + 4.85·53-s − 1.14·59-s + 0.854·61-s + 7·67-s − 9·71-s − 2.70·73-s − 5.29·77-s + 11.7·79-s + 6.70·83-s + 12·89-s − 2.43·91-s + 10·97-s − 4.85·101-s − 0.145·103-s + ⋯ |
| L(s) = 1 | + 1.07·7-s − 0.559·11-s − 0.236·13-s + 0.277·17-s + 0.458·19-s − 1.01·23-s − 0.688·29-s + 0.486·31-s + 0.962·37-s + 1.80·41-s − 0.130·43-s − 0.978·47-s + 0.163·49-s + 0.666·53-s − 0.149·59-s + 0.109·61-s + 0.855·67-s − 1.06·71-s − 0.316·73-s − 0.603·77-s + 1.31·79-s + 0.736·83-s + 1.27·89-s − 0.255·91-s + 1.01·97-s − 0.483·101-s − 0.0143·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.205359796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.205359796\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2.85T + 7T^{2} \) |
| 11 | \( 1 + 1.85T + 11T^{2} \) |
| 13 | \( 1 + 0.854T + 13T^{2} \) |
| 17 | \( 1 - 1.14T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 4.85T + 23T^{2} \) |
| 29 | \( 1 + 3.70T + 29T^{2} \) |
| 31 | \( 1 - 2.70T + 31T^{2} \) |
| 37 | \( 1 - 5.85T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 + 0.854T + 43T^{2} \) |
| 47 | \( 1 + 6.70T + 47T^{2} \) |
| 53 | \( 1 - 4.85T + 53T^{2} \) |
| 59 | \( 1 + 1.14T + 59T^{2} \) |
| 61 | \( 1 - 0.854T + 61T^{2} \) |
| 67 | \( 1 - 7T + 67T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 + 2.70T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 - 6.70T + 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−7.77184137464244897689466070234, −7.43134629997727449236896820539, −6.33958438870765550188289106726, −5.71220077342642415915387217532, −4.98746182925044373275434882026, −4.40724150083215589336753543905, −3.54217383742121329028495354525, −2.54512684587803881689473192536, −1.82920961333447795625403777510, −0.74089191465866742623027183535,
0.74089191465866742623027183535, 1.82920961333447795625403777510, 2.54512684587803881689473192536, 3.54217383742121329028495354525, 4.40724150083215589336753543905, 4.98746182925044373275434882026, 5.71220077342642415915387217532, 6.33958438870765550188289106726, 7.43134629997727449236896820539, 7.77184137464244897689466070234
