L(s) = 1 | − 6.91e3·3-s − 2.45e5·5-s − 3.75e5·7-s + 3.34e7·9-s − 7.40e7·11-s + 2.38e8·13-s + 1.69e9·15-s + 7.12e8·17-s − 1.63e9·19-s + 2.59e9·21-s − 1.84e10·23-s + 2.98e10·25-s − 1.31e11·27-s − 7.74e10·29-s + 1.32e11·31-s + 5.11e11·33-s + 9.22e10·35-s + 1.16e11·37-s − 1.64e12·39-s + 2.32e12·41-s + 2.08e12·43-s − 8.20e12·45-s + 2.33e12·47-s − 4.60e12·49-s − 4.92e12·51-s + 6.30e12·53-s + 1.81e13·55-s + ⋯ |
L(s) = 1 | − 1.82·3-s − 1.40·5-s − 0.172·7-s + 2.32·9-s − 1.14·11-s + 1.05·13-s + 2.56·15-s + 0.421·17-s − 0.418·19-s + 0.314·21-s − 1.12·23-s + 0.976·25-s − 2.42·27-s − 0.833·29-s + 0.868·31-s + 2.08·33-s + 0.242·35-s + 0.202·37-s − 1.92·39-s + 1.86·41-s + 1.17·43-s − 3.27·45-s + 0.673·47-s − 0.970·49-s − 0.768·51-s + 0.737·53-s + 1.60·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(8)\) |
\(\approx\) |
\(0.4575655979\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4575655979\) |
\(L(\frac{17}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + 6.91e3T + 1.43e7T^{2} \) |
| 5 | \( 1 + 2.45e5T + 3.05e10T^{2} \) |
| 7 | \( 1 + 3.75e5T + 4.74e12T^{2} \) |
| 11 | \( 1 + 7.40e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 2.38e8T + 5.11e16T^{2} \) |
| 17 | \( 1 - 7.12e8T + 2.86e18T^{2} \) |
| 19 | \( 1 + 1.63e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + 1.84e10T + 2.66e20T^{2} \) |
| 29 | \( 1 + 7.74e10T + 8.62e21T^{2} \) |
| 31 | \( 1 - 1.32e11T + 2.34e22T^{2} \) |
| 37 | \( 1 - 1.16e11T + 3.33e23T^{2} \) |
| 41 | \( 1 - 2.32e12T + 1.55e24T^{2} \) |
| 43 | \( 1 - 2.08e12T + 3.17e24T^{2} \) |
| 47 | \( 1 - 2.33e12T + 1.20e25T^{2} \) |
| 53 | \( 1 - 6.30e12T + 7.31e25T^{2} \) |
| 59 | \( 1 - 2.90e12T + 3.65e26T^{2} \) |
| 61 | \( 1 - 2.66e12T + 6.02e26T^{2} \) |
| 67 | \( 1 - 1.67e12T + 2.46e27T^{2} \) |
| 71 | \( 1 + 8.11e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + 9.97e13T + 8.90e27T^{2} \) |
| 79 | \( 1 - 7.04e13T + 2.91e28T^{2} \) |
| 83 | \( 1 - 2.99e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + 6.60e14T + 1.74e29T^{2} \) |
| 97 | \( 1 - 1.46e15T + 6.33e29T^{2} \) |
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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
−17.93141819570888901536804707999, −16.28672563384899712417247302567, −15.66908827177854488616155138602, −12.77475607150245383652151371922, −11.59420983772793419292327645457, −10.57462376958295016153351411067, −7.67793198394861137419437457875, −5.90039996848995902934037329616, −4.19861867695133090937414233451, −0.56824712043044151201863432086,
0.56824712043044151201863432086, 4.19861867695133090937414233451, 5.90039996848995902934037329616, 7.67793198394861137419437457875, 10.57462376958295016153351411067, 11.59420983772793419292327645457, 12.77475607150245383652151371922, 15.66908827177854488616155138602, 16.28672563384899712417247302567, 17.93141819570888901536804707999