| L(s) = 1 | + 2.82·5-s + 2·11-s − 5.65·13-s + 2.82·17-s + 5.65·19-s + 6·23-s + 3.00·25-s + 4·29-s − 5.65·31-s − 2·37-s − 2.82·41-s + 4·43-s + 11.3·47-s − 12·53-s + 5.65·55-s + 11.3·59-s − 5.65·61-s − 16.0·65-s + 12·67-s − 6·71-s − 8·79-s − 11.3·83-s + 8.00·85-s + 8.48·89-s + 16.0·95-s + 11.3·97-s + 8.48·101-s + ⋯ |
| L(s) = 1 | + 1.26·5-s + 0.603·11-s − 1.56·13-s + 0.685·17-s + 1.29·19-s + 1.25·23-s + 0.600·25-s + 0.742·29-s − 1.01·31-s − 0.328·37-s − 0.441·41-s + 0.609·43-s + 1.65·47-s − 1.64·53-s + 0.762·55-s + 1.47·59-s − 0.724·61-s − 1.98·65-s + 1.46·67-s − 0.712·71-s − 0.900·79-s − 1.24·83-s + 0.867·85-s + 0.899·89-s + 1.64·95-s + 1.14·97-s + 0.844·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.831736179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.831736179\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 2.82T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 8.48T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−7.73444348375358982236853316986, −7.22950915105850518195531771543, −6.59554341833355144531050958602, −5.65472479360835775346575654376, −5.28484912889112647816858318845, −4.54711827913441491712585287685, −3.38364402230476885099865176310, −2.67938830523706588195711504339, −1.81984183259206743301132092386, −0.887945462437249440146770848193,
0.887945462437249440146770848193, 1.81984183259206743301132092386, 2.67938830523706588195711504339, 3.38364402230476885099865176310, 4.54711827913441491712585287685, 5.28484912889112647816858318845, 5.65472479360835775346575654376, 6.59554341833355144531050958602, 7.22950915105850518195531771543, 7.73444348375358982236853316986
