| L(s) = 1 | − 2.32·2-s + 2.41·3-s + 3.39·4-s + 0.249·5-s − 5.61·6-s + 7-s − 3.24·8-s + 2.83·9-s − 0.580·10-s + 3.91·11-s + 8.20·12-s + 0.778·13-s − 2.32·14-s + 0.603·15-s + 0.747·16-s + 3.22·17-s − 6.58·18-s + 6.57·19-s + 0.849·20-s + 2.41·21-s − 9.09·22-s − 3.71·23-s − 7.84·24-s − 4.93·25-s − 1.80·26-s − 0.394·27-s + 3.39·28-s + ⋯ |
| L(s) = 1 | − 1.64·2-s + 1.39·3-s + 1.69·4-s + 0.111·5-s − 2.29·6-s + 0.377·7-s − 1.14·8-s + 0.945·9-s − 0.183·10-s + 1.18·11-s + 2.36·12-s + 0.215·13-s − 0.620·14-s + 0.155·15-s + 0.186·16-s + 0.782·17-s − 1.55·18-s + 1.50·19-s + 0.189·20-s + 0.527·21-s − 1.93·22-s − 0.773·23-s − 1.60·24-s − 0.987·25-s − 0.354·26-s − 0.0759·27-s + 0.642·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.991952320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.991952320\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 863 | \( 1 + T \) |
| good | 2 | \( 1 + 2.32T + 2T^{2} \) |
| 3 | \( 1 - 2.41T + 3T^{2} \) |
| 5 | \( 1 - 0.249T + 5T^{2} \) |
| 11 | \( 1 - 3.91T + 11T^{2} \) |
| 13 | \( 1 - 0.778T + 13T^{2} \) |
| 17 | \( 1 - 3.22T + 17T^{2} \) |
| 19 | \( 1 - 6.57T + 19T^{2} \) |
| 23 | \( 1 + 3.71T + 23T^{2} \) |
| 29 | \( 1 - 9.52T + 29T^{2} \) |
| 31 | \( 1 - 7.29T + 31T^{2} \) |
| 37 | \( 1 + 8.87T + 37T^{2} \) |
| 41 | \( 1 - 4.01T + 41T^{2} \) |
| 43 | \( 1 - 2.12T + 43T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 - 4.92T + 53T^{2} \) |
| 59 | \( 1 + 3.00T + 59T^{2} \) |
| 61 | \( 1 - 7.07T + 61T^{2} \) |
| 67 | \( 1 - 9.24T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 3.92T + 73T^{2} \) |
| 79 | \( 1 - 9.95T + 79T^{2} \) |
| 83 | \( 1 + 17.4T + 83T^{2} \) |
| 89 | \( 1 + 1.86T + 89T^{2} \) |
| 97 | \( 1 - 3.66T + 97T^{2} \) |
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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
−8.068228257208612538010211332503, −7.934530823734633186479208104198, −6.96570213323416208107475241663, −6.41775164195262308546349895069, −5.29226113665982974372723080587, −4.11921409154077524022175488727, −3.34641219597273635407845983082, −2.51693366796927985655665912656, −1.62610532023704685399921312537, −0.977288652753474133229436819361,
0.977288652753474133229436819361, 1.62610532023704685399921312537, 2.51693366796927985655665912656, 3.34641219597273635407845983082, 4.11921409154077524022175488727, 5.29226113665982974372723080587, 6.41775164195262308546349895069, 6.96570213323416208107475241663, 7.934530823734633186479208104198, 8.068228257208612538010211332503
