L(s) = 1 | − 0.962·3-s + 2.73·5-s − 4.42·7-s − 2.07·9-s − 2.59·11-s + 2.44·13-s − 2.62·15-s + 0.359·17-s + 4.11·19-s + 4.26·21-s + 0.0487·23-s + 2.45·25-s + 4.88·27-s + 1.62·29-s − 1.82·31-s + 2.49·33-s − 12.0·35-s + 7.04·37-s − 2.35·39-s − 2.33·41-s − 0.598·43-s − 5.66·45-s + 12.9·47-s + 12.5·49-s − 0.345·51-s + 3.71·53-s − 7.07·55-s + ⋯ |
L(s) = 1 | − 0.555·3-s + 1.22·5-s − 1.67·7-s − 0.691·9-s − 0.780·11-s + 0.677·13-s − 0.678·15-s + 0.0870·17-s + 0.944·19-s + 0.929·21-s + 0.0101·23-s + 0.491·25-s + 0.939·27-s + 0.302·29-s − 0.327·31-s + 0.434·33-s − 2.04·35-s + 1.15·37-s − 0.376·39-s − 0.364·41-s − 0.0913·43-s − 0.843·45-s + 1.89·47-s + 1.79·49-s − 0.0483·51-s + 0.510·53-s − 0.953·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 + T \) |
good | 3 | \( 1 + 0.962T + 3T^{2} \) |
| 5 | \( 1 - 2.73T + 5T^{2} \) |
| 7 | \( 1 + 4.42T + 7T^{2} \) |
| 11 | \( 1 + 2.59T + 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 - 0.359T + 17T^{2} \) |
| 19 | \( 1 - 4.11T + 19T^{2} \) |
| 23 | \( 1 - 0.0487T + 23T^{2} \) |
| 29 | \( 1 - 1.62T + 29T^{2} \) |
| 31 | \( 1 + 1.82T + 31T^{2} \) |
| 37 | \( 1 - 7.04T + 37T^{2} \) |
| 41 | \( 1 + 2.33T + 41T^{2} \) |
| 43 | \( 1 + 0.598T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 - 3.71T + 53T^{2} \) |
| 59 | \( 1 - 0.779T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 + 5.58T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 + 5.23T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 6.27T + 97T^{2} \) |
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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
−7.60160686255466022124076306121, −6.78538182163646109617514121668, −6.07312414120916492824183547425, −5.79284317264107314301580692691, −5.19158291961321740470336378628, −3.98123133970416205239696050648, −2.95645840734894056504325166317, −2.59963359471257012910610964354, −1.18154424234106845908492046374, 0,
1.18154424234106845908492046374, 2.59963359471257012910610964354, 2.95645840734894056504325166317, 3.98123133970416205239696050648, 5.19158291961321740470336378628, 5.79284317264107314301580692691, 6.07312414120916492824183547425, 6.78538182163646109617514121668, 7.60160686255466022124076306121