L(s) = 1 | − 3.35·3-s − 0.581·5-s + 3.25·7-s + 8.27·9-s + 4.01·11-s + 4.26·13-s + 1.95·15-s − 2.04·17-s + 19-s − 10.9·21-s + 0.293·23-s − 4.66·25-s − 17.7·27-s − 6.94·29-s − 1.87·31-s − 13.4·33-s − 1.89·35-s + 8.28·37-s − 14.3·39-s − 6.21·41-s − 9.11·43-s − 4.81·45-s − 8.53·47-s + 3.57·49-s + 6.87·51-s − 14.3·53-s − 2.33·55-s + ⋯ |
L(s) = 1 | − 1.93·3-s − 0.259·5-s + 1.22·7-s + 2.75·9-s + 1.21·11-s + 1.18·13-s + 0.504·15-s − 0.496·17-s + 0.229·19-s − 2.38·21-s + 0.0611·23-s − 0.932·25-s − 3.41·27-s − 1.28·29-s − 0.336·31-s − 2.34·33-s − 0.319·35-s + 1.36·37-s − 2.29·39-s − 0.969·41-s − 1.38·43-s − 0.717·45-s − 1.24·47-s + 0.511·49-s + 0.962·51-s − 1.96·53-s − 0.315·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 + T \) |
good | 3 | \( 1 + 3.35T + 3T^{2} \) |
| 5 | \( 1 + 0.581T + 5T^{2} \) |
| 7 | \( 1 - 3.25T + 7T^{2} \) |
| 11 | \( 1 - 4.01T + 11T^{2} \) |
| 13 | \( 1 - 4.26T + 13T^{2} \) |
| 17 | \( 1 + 2.04T + 17T^{2} \) |
| 23 | \( 1 - 0.293T + 23T^{2} \) |
| 29 | \( 1 + 6.94T + 29T^{2} \) |
| 31 | \( 1 + 1.87T + 31T^{2} \) |
| 37 | \( 1 - 8.28T + 37T^{2} \) |
| 41 | \( 1 + 6.21T + 41T^{2} \) |
| 43 | \( 1 + 9.11T + 43T^{2} \) |
| 47 | \( 1 + 8.53T + 47T^{2} \) |
| 53 | \( 1 + 14.3T + 53T^{2} \) |
| 59 | \( 1 - 3.68T + 59T^{2} \) |
| 61 | \( 1 - 1.48T + 61T^{2} \) |
| 67 | \( 1 - 2.71T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 - 9.47T + 73T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 7.66T + 89T^{2} \) |
| 97 | \( 1 + 18.9T + 97T^{2} \) |
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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
−7.66348823356654109181237201338, −6.66724061964615445019243928828, −6.38551234582863626687055597107, −5.53722333953794255936352928060, −4.97166108172107545906692090002, −4.19651990851201108423673107156, −3.69656708894969024374859774750, −1.66201084311837848829378398419, −1.35656518152428276287307997563, 0,
1.35656518152428276287307997563, 1.66201084311837848829378398419, 3.69656708894969024374859774750, 4.19651990851201108423673107156, 4.97166108172107545906692090002, 5.53722333953794255936352928060, 6.38551234582863626687055597107, 6.66724061964615445019243928828, 7.66348823356654109181237201338