L(s) = 1 | − 64·2-s + 2.22e3·3-s + 4.09e3·4-s − 3.19e4·5-s − 1.42e5·6-s + 3.02e5·7-s − 2.62e5·8-s + 3.35e6·9-s + 2.04e6·10-s + 1.77e6·11-s + 9.11e6·12-s + 2.00e6·13-s − 1.93e7·14-s − 7.11e7·15-s + 1.67e7·16-s + 5.06e7·17-s − 2.15e8·18-s − 1.58e8·19-s − 1.30e8·20-s + 6.74e8·21-s − 1.13e8·22-s + 9.79e8·23-s − 5.83e8·24-s − 1.99e8·25-s − 1.28e8·26-s + 3.92e9·27-s + 1.24e9·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.76·3-s + 0.5·4-s − 0.914·5-s − 1.24·6-s + 0.973·7-s − 0.353·8-s + 2.10·9-s + 0.646·10-s + 0.301·11-s + 0.881·12-s + 0.115·13-s − 0.688·14-s − 1.61·15-s + 0.250·16-s + 0.508·17-s − 1.49·18-s − 0.772·19-s − 0.457·20-s + 1.71·21-s − 0.213·22-s + 1.38·23-s − 0.623·24-s − 0.163·25-s − 0.0814·26-s + 1.95·27-s + 0.486·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(7)\) |
\(\approx\) |
\(2.639071398\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.639071398\) |
\(L(\frac{15}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 64T \) |
| 11 | \( 1 - 1.77e6T \) |
good | 3 | \( 1 - 2.22e3T + 1.59e6T^{2} \) |
| 5 | \( 1 + 3.19e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 3.02e5T + 9.68e10T^{2} \) |
| 13 | \( 1 - 2.00e6T + 3.02e14T^{2} \) |
| 17 | \( 1 - 5.06e7T + 9.90e15T^{2} \) |
| 19 | \( 1 + 1.58e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 9.79e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 5.86e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 1.21e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 2.66e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 4.14e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 5.82e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 1.16e11T + 5.46e21T^{2} \) |
| 53 | \( 1 - 4.45e10T + 2.60e22T^{2} \) |
| 59 | \( 1 + 6.41e10T + 1.04e23T^{2} \) |
| 61 | \( 1 + 4.11e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 3.99e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.61e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + 2.11e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 2.00e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 1.16e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 5.55e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 9.32e12T + 6.73e25T^{2} \) |
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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
−14.95140131802103981201867073410, −13.99997988268235395646032530887, −12.25823527277493036412370418465, −10.66542434971115677112512000047, −9.013362933866213392521717675652, −8.217711651106328436347817370147, −7.27939813540220854995296396707, −4.21473712046343081174677530835, −2.75707556591662842592296823267, −1.24729011454642587542203679716,
1.24729011454642587542203679716, 2.75707556591662842592296823267, 4.21473712046343081174677530835, 7.27939813540220854995296396707, 8.217711651106328436347817370147, 9.013362933866213392521717675652, 10.66542434971115677112512000047, 12.25823527277493036412370418465, 13.99997988268235395646032530887, 14.95140131802103981201867073410