| L(s) = 1 | + 0.851·3-s + 4.11·5-s − 4.22·7-s − 2.27·9-s + 4.95·11-s + 3.50·15-s − 4.42·17-s + 3.17·19-s − 3.59·21-s + 3.80·23-s + 11.9·25-s − 4.49·27-s − 1.10·29-s + 7.19·31-s + 4.22·33-s − 17.3·35-s + 0.218·37-s − 9.16·41-s + 11.1·43-s − 9.37·45-s + 4.49·47-s + 10.8·49-s − 3.76·51-s + 4.87·53-s + 20.4·55-s + 2.70·57-s + 0.982·59-s + ⋯ |
| L(s) = 1 | + 0.491·3-s + 1.84·5-s − 1.59·7-s − 0.758·9-s + 1.49·11-s + 0.905·15-s − 1.07·17-s + 0.728·19-s − 0.784·21-s + 0.794·23-s + 2.39·25-s − 0.864·27-s − 0.205·29-s + 1.29·31-s + 0.735·33-s − 2.94·35-s + 0.0358·37-s − 1.43·41-s + 1.70·43-s − 1.39·45-s + 0.655·47-s + 1.54·49-s − 0.527·51-s + 0.668·53-s + 2.75·55-s + 0.358·57-s + 0.127·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.949571778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.949571778\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 0.851T + 3T^{2} \) |
| 5 | \( 1 - 4.11T + 5T^{2} \) |
| 7 | \( 1 + 4.22T + 7T^{2} \) |
| 11 | \( 1 - 4.95T + 11T^{2} \) |
| 17 | \( 1 + 4.42T + 17T^{2} \) |
| 19 | \( 1 - 3.17T + 19T^{2} \) |
| 23 | \( 1 - 3.80T + 23T^{2} \) |
| 29 | \( 1 + 1.10T + 29T^{2} \) |
| 31 | \( 1 - 7.19T + 31T^{2} \) |
| 37 | \( 1 - 0.218T + 37T^{2} \) |
| 41 | \( 1 + 9.16T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 4.49T + 47T^{2} \) |
| 53 | \( 1 - 4.87T + 53T^{2} \) |
| 59 | \( 1 - 0.982T + 59T^{2} \) |
| 61 | \( 1 + 6.00T + 61T^{2} \) |
| 67 | \( 1 - 3.15T + 67T^{2} \) |
| 71 | \( 1 - 5.39T + 71T^{2} \) |
| 73 | \( 1 + 5.36T + 73T^{2} \) |
| 79 | \( 1 - 1.04T + 79T^{2} \) |
| 83 | \( 1 + 7.56T + 83T^{2} \) |
| 89 | \( 1 + 6.65T + 89T^{2} \) |
| 97 | \( 1 - 1.42T + 97T^{2} \) |
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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
−8.601156833556075902669010836475, −7.12183875586457856733166441307, −6.64541787302737795728292223770, −6.06898551002428389228901966980, −5.59332219004030232706119863005, −4.47606516885810095364209315545, −3.40082855238052155088605922714, −2.80953374342725854345360867399, −2.06371453277617785881014055622, −0.910725170294548524064404527871,
0.910725170294548524064404527871, 2.06371453277617785881014055622, 2.80953374342725854345360867399, 3.40082855238052155088605922714, 4.47606516885810095364209315545, 5.59332219004030232706119863005, 6.06898551002428389228901966980, 6.64541787302737795728292223770, 7.12183875586457856733166441307, 8.601156833556075902669010836475
