| L(s) = 1 | + 2.37·2-s + 3.66·4-s − 2.66·5-s + 3.95·8-s − 6.33·10-s − 1.57·11-s − 13-s + 2.08·16-s + 4.75·17-s + 2.23·19-s − 9.74·20-s − 3.74·22-s − 5.84·23-s + 2.08·25-s − 2.37·26-s − 4.23·29-s − 7.28·31-s − 2.94·32-s + 11.3·34-s + 10.4·37-s + 5.32·38-s − 10.5·40-s − 2.25·41-s + 0.913·43-s − 5.76·44-s − 13.9·46-s − 2.09·47-s + ⋯ |
| L(s) = 1 | + 1.68·2-s + 1.83·4-s − 1.19·5-s + 1.39·8-s − 2.00·10-s − 0.475·11-s − 0.277·13-s + 0.521·16-s + 1.15·17-s + 0.513·19-s − 2.17·20-s − 0.799·22-s − 1.21·23-s + 0.417·25-s − 0.466·26-s − 0.786·29-s − 1.30·31-s − 0.520·32-s + 1.94·34-s + 1.72·37-s + 0.863·38-s − 1.66·40-s − 0.351·41-s + 0.139·43-s − 0.869·44-s − 2.05·46-s − 0.305·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 2.37T + 2T^{2} \) |
| 5 | \( 1 + 2.66T + 5T^{2} \) |
| 11 | \( 1 + 1.57T + 11T^{2} \) |
| 17 | \( 1 - 4.75T + 17T^{2} \) |
| 19 | \( 1 - 2.23T + 19T^{2} \) |
| 23 | \( 1 + 5.84T + 23T^{2} \) |
| 29 | \( 1 + 4.23T + 29T^{2} \) |
| 31 | \( 1 + 7.28T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 2.25T + 41T^{2} \) |
| 43 | \( 1 - 0.913T + 43T^{2} \) |
| 47 | \( 1 + 2.09T + 47T^{2} \) |
| 53 | \( 1 + 1.08T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 - 7.51T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 7.42T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−7.58564101034084587574991872705, −7.09535393933592638165982098691, −5.86911447022884152008077434673, −5.69209077603490395745469986542, −4.64441805420676099603666516172, −4.15250843735192242550808251397, −3.40650829547879088130661958246, −2.85489606518411439928105635833, −1.68795370205762329259032125000, 0,
1.68795370205762329259032125000, 2.85489606518411439928105635833, 3.40650829547879088130661958246, 4.15250843735192242550808251397, 4.64441805420676099603666516172, 5.69209077603490395745469986542, 5.86911447022884152008077434673, 7.09535393933592638165982098691, 7.58564101034084587574991872705
