| L(s) = 1 | + 1.48·3-s + 1.28·5-s + 0.0101·7-s − 0.786·9-s − 2.81·11-s − 4.93·13-s + 1.91·15-s + 17-s + 0.137·19-s + 0.0151·21-s + 2.88·23-s − 3.34·25-s − 5.63·27-s + 0.119·29-s + 1.47·31-s − 4.19·33-s + 0.0130·35-s − 7.73·37-s − 7.34·39-s + 0.257·41-s + 3.65·43-s − 1.01·45-s + 4.25·47-s − 6.99·49-s + 1.48·51-s + 7.93·53-s − 3.62·55-s + ⋯ |
| L(s) = 1 | + 0.858·3-s + 0.574·5-s + 0.00384·7-s − 0.262·9-s − 0.849·11-s − 1.36·13-s + 0.493·15-s + 0.242·17-s + 0.0316·19-s + 0.00329·21-s + 0.602·23-s − 0.669·25-s − 1.08·27-s + 0.0222·29-s + 0.264·31-s − 0.730·33-s + 0.00220·35-s − 1.27·37-s − 1.17·39-s + 0.0401·41-s + 0.557·43-s − 0.150·45-s + 0.620·47-s − 0.999·49-s + 0.208·51-s + 1.08·53-s − 0.488·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 1.48T + 3T^{2} \) |
| 5 | \( 1 - 1.28T + 5T^{2} \) |
| 7 | \( 1 - 0.0101T + 7T^{2} \) |
| 11 | \( 1 + 2.81T + 11T^{2} \) |
| 13 | \( 1 + 4.93T + 13T^{2} \) |
| 19 | \( 1 - 0.137T + 19T^{2} \) |
| 23 | \( 1 - 2.88T + 23T^{2} \) |
| 29 | \( 1 - 0.119T + 29T^{2} \) |
| 31 | \( 1 - 1.47T + 31T^{2} \) |
| 37 | \( 1 + 7.73T + 37T^{2} \) |
| 41 | \( 1 - 0.257T + 41T^{2} \) |
| 43 | \( 1 - 3.65T + 43T^{2} \) |
| 47 | \( 1 - 4.25T + 47T^{2} \) |
| 53 | \( 1 - 7.93T + 53T^{2} \) |
| 61 | \( 1 + 15.4T + 61T^{2} \) |
| 67 | \( 1 + 9.20T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 - 7.46T + 73T^{2} \) |
| 79 | \( 1 + 8.24T + 79T^{2} \) |
| 83 | \( 1 - 8.73T + 83T^{2} \) |
| 89 | \( 1 + 7.42T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−7.924144088864609010607878856654, −7.62325041198209233814868066723, −6.72157478881348705832880381604, −5.69055626979587762191035016886, −5.19389757075056482715666319652, −4.24255187592165428544081077402, −3.08415434663906859485257883771, −2.60464482561605444619438728396, −1.72961397883709766709091805814, 0,
1.72961397883709766709091805814, 2.60464482561605444619438728396, 3.08415434663906859485257883771, 4.24255187592165428544081077402, 5.19389757075056482715666319652, 5.69055626979587762191035016886, 6.72157478881348705832880381604, 7.62325041198209233814868066723, 7.924144088864609010607878856654
