| L(s) = 1 | − 2-s + 2.61·3-s + 4-s − 5-s − 2.61·6-s + 1.61·7-s − 8-s + 3.85·9-s + 10-s + 4.47·11-s + 2.61·12-s + 1.85·13-s − 1.61·14-s − 2.61·15-s + 16-s − 1.61·17-s − 3.85·18-s + 3.23·19-s − 20-s + 4.23·21-s − 4.47·22-s + 2.61·23-s − 2.61·24-s + 25-s − 1.85·26-s + 2.23·27-s + 1.61·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.51·3-s + 0.5·4-s − 0.447·5-s − 1.06·6-s + 0.611·7-s − 0.353·8-s + 1.28·9-s + 0.316·10-s + 1.34·11-s + 0.755·12-s + 0.514·13-s − 0.432·14-s − 0.675·15-s + 0.250·16-s − 0.392·17-s − 0.908·18-s + 0.742·19-s − 0.223·20-s + 0.924·21-s − 0.953·22-s + 0.545·23-s − 0.534·24-s + 0.200·25-s − 0.363·26-s + 0.430·27-s + 0.305·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.209590001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.209590001\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - 2.61T + 3T^{2} \) |
| 7 | \( 1 - 1.61T + 7T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 - 1.85T + 13T^{2} \) |
| 17 | \( 1 + 1.61T + 17T^{2} \) |
| 19 | \( 1 - 3.23T + 19T^{2} \) |
| 23 | \( 1 - 2.61T + 23T^{2} \) |
| 31 | \( 1 - 0.854T + 31T^{2} \) |
| 37 | \( 1 - 2.47T + 37T^{2} \) |
| 41 | \( 1 - 7.23T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 - 9.70T + 47T^{2} \) |
| 53 | \( 1 + 2.38T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 3.09T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 + 16.4T + 71T^{2} \) |
| 73 | \( 1 + 3.14T + 73T^{2} \) |
| 79 | \( 1 - 0.145T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 - 5.56T + 97T^{2} \) |
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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
−8.010524807594593850657418476428, −7.31011409409926424058452793219, −6.79668217183013345467743909294, −5.91269913380362854631821728916, −4.78514009825110036677027608934, −3.97059671078500396056025895027, −3.42266393765781111276499060399, −2.60349215121403379257722330520, −1.70798692789326673227518689646, −0.983916242204349899688750483846,
0.983916242204349899688750483846, 1.70798692789326673227518689646, 2.60349215121403379257722330520, 3.42266393765781111276499060399, 3.97059671078500396056025895027, 4.78514009825110036677027608934, 5.91269913380362854631821728916, 6.79668217183013345467743909294, 7.31011409409926424058452793219, 8.010524807594593850657418476428
