| L(s) = 1 | + 2.93·3-s + 3.18·5-s − 3.42·7-s + 5.61·9-s + 4.68·11-s − 2.68·13-s + 9.36·15-s − 1.49·17-s + 1.25·19-s − 10.0·21-s − 5.10·23-s + 5.17·25-s + 7.68·27-s − 5.12·29-s + 8.29·31-s + 13.7·33-s − 10.9·35-s + 5.95·37-s − 7.87·39-s − 4.63·41-s − 5.01·43-s + 17.9·45-s + 8.37·47-s + 4.74·49-s − 4.37·51-s − 14.0·53-s + 14.9·55-s + ⋯ |
| L(s) = 1 | + 1.69·3-s + 1.42·5-s − 1.29·7-s + 1.87·9-s + 1.41·11-s − 0.743·13-s + 2.41·15-s − 0.361·17-s + 0.287·19-s − 2.19·21-s − 1.06·23-s + 1.03·25-s + 1.47·27-s − 0.951·29-s + 1.49·31-s + 2.39·33-s − 1.84·35-s + 0.978·37-s − 1.26·39-s − 0.723·41-s − 0.764·43-s + 2.67·45-s + 1.22·47-s + 0.677·49-s − 0.613·51-s − 1.92·53-s + 2.01·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.246067120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.246067120\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 3 | \( 1 - 2.93T + 3T^{2} \) |
| 5 | \( 1 - 3.18T + 5T^{2} \) |
| 7 | \( 1 + 3.42T + 7T^{2} \) |
| 11 | \( 1 - 4.68T + 11T^{2} \) |
| 13 | \( 1 + 2.68T + 13T^{2} \) |
| 17 | \( 1 + 1.49T + 17T^{2} \) |
| 19 | \( 1 - 1.25T + 19T^{2} \) |
| 23 | \( 1 + 5.10T + 23T^{2} \) |
| 29 | \( 1 + 5.12T + 29T^{2} \) |
| 31 | \( 1 - 8.29T + 31T^{2} \) |
| 37 | \( 1 - 5.95T + 37T^{2} \) |
| 41 | \( 1 + 4.63T + 41T^{2} \) |
| 43 | \( 1 + 5.01T + 43T^{2} \) |
| 47 | \( 1 - 8.37T + 47T^{2} \) |
| 53 | \( 1 + 14.0T + 53T^{2} \) |
| 59 | \( 1 - 4.81T + 59T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 8.23T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 9.56T + 79T^{2} \) |
| 83 | \( 1 + 5.47T + 83T^{2} \) |
| 89 | \( 1 - 2.88T + 89T^{2} \) |
| 97 | \( 1 - 0.427T + 97T^{2} \) |
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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
−9.724749680110405700151888377619, −9.341720257344709274202304717311, −8.646939037591302329662872948426, −7.51622842482755747363817164005, −6.59133483941802517253018307417, −6.00311073424424769779512946268, −4.43567660601429954047817760853, −3.42081281079598256609411341634, −2.57439834850206064860367208303, −1.66121487567519946478556611837,
1.66121487567519946478556611837, 2.57439834850206064860367208303, 3.42081281079598256609411341634, 4.43567660601429954047817760853, 6.00311073424424769779512946268, 6.59133483941802517253018307417, 7.51622842482755747363817164005, 8.646939037591302329662872948426, 9.341720257344709274202304717311, 9.724749680110405700151888377619
