| L(s) = 1 | + 1.34·2-s − 0.184·4-s − 2.53·5-s − 2.94·8-s − 3.41·10-s + 0.467·11-s − 5.82·13-s − 3.59·16-s − 3.87·17-s + 2.18·19-s + 0.467·20-s + 0.630·22-s − 0.106·23-s + 1.41·25-s − 7.84·26-s + 8.78·29-s + 7.68·31-s + 1.04·32-s − 5.22·34-s − 7.68·37-s + 2.94·38-s + 7.45·40-s + 2.22·41-s + 1.22·43-s − 0.0864·44-s − 0.142·46-s + 5.33·47-s + ⋯ |
| L(s) = 1 | + 0.952·2-s − 0.0923·4-s − 1.13·5-s − 1.04·8-s − 1.07·10-s + 0.141·11-s − 1.61·13-s − 0.899·16-s − 0.940·17-s + 0.501·19-s + 0.104·20-s + 0.134·22-s − 0.0221·23-s + 0.282·25-s − 1.53·26-s + 1.63·29-s + 1.37·31-s + 0.184·32-s − 0.896·34-s − 1.26·37-s + 0.477·38-s + 1.17·40-s + 0.347·41-s + 0.187·43-s − 0.0130·44-s − 0.0210·46-s + 0.777·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.329253637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.329253637\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.34T + 2T^{2} \) |
| 5 | \( 1 + 2.53T + 5T^{2} \) |
| 11 | \( 1 - 0.467T + 11T^{2} \) |
| 13 | \( 1 + 5.82T + 13T^{2} \) |
| 17 | \( 1 + 3.87T + 17T^{2} \) |
| 19 | \( 1 - 2.18T + 19T^{2} \) |
| 23 | \( 1 + 0.106T + 23T^{2} \) |
| 29 | \( 1 - 8.78T + 29T^{2} \) |
| 31 | \( 1 - 7.68T + 31T^{2} \) |
| 37 | \( 1 + 7.68T + 37T^{2} \) |
| 41 | \( 1 - 2.22T + 41T^{2} \) |
| 43 | \( 1 - 1.22T + 43T^{2} \) |
| 47 | \( 1 - 5.33T + 47T^{2} \) |
| 53 | \( 1 + 0.716T + 53T^{2} \) |
| 59 | \( 1 + 0.736T + 59T^{2} \) |
| 61 | \( 1 + 0.958T + 61T^{2} \) |
| 67 | \( 1 + 9.63T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 2.73T + 83T^{2} \) |
| 89 | \( 1 - 8.11T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−8.403551836718568435325286579530, −7.65343054389167251710874939906, −6.91434687199022591079729091594, −6.22121959168113238585658593585, −5.09698755810419350901288054824, −4.66901922864519360430678073066, −4.04114773342861518039251386124, −3.13877817271932927446041297401, −2.40193234121672161280718253935, −0.54815999522225199395145748178,
0.54815999522225199395145748178, 2.40193234121672161280718253935, 3.13877817271932927446041297401, 4.04114773342861518039251386124, 4.66901922864519360430678073066, 5.09698755810419350901288054824, 6.22121959168113238585658593585, 6.91434687199022591079729091594, 7.65343054389167251710874939906, 8.403551836718568435325286579530
