| L(s) = 1 | + 2-s + 1.76·3-s + 4-s − 3.63·5-s + 1.76·6-s − 3.98·7-s + 8-s + 0.103·9-s − 3.63·10-s − 11-s + 1.76·12-s + 6.84·13-s − 3.98·14-s − 6.39·15-s + 16-s − 7.57·17-s + 0.103·18-s + 6.99·19-s − 3.63·20-s − 7.01·21-s − 22-s + 7.14·23-s + 1.76·24-s + 8.19·25-s + 6.84·26-s − 5.10·27-s − 3.98·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.01·3-s + 0.5·4-s − 1.62·5-s + 0.719·6-s − 1.50·7-s + 0.353·8-s + 0.0345·9-s − 1.14·10-s − 0.301·11-s + 0.508·12-s + 1.89·13-s − 1.06·14-s − 1.65·15-s + 0.250·16-s − 1.83·17-s + 0.0244·18-s + 1.60·19-s − 0.812·20-s − 1.53·21-s − 0.213·22-s + 1.48·23-s + 0.359·24-s + 1.63·25-s + 1.34·26-s − 0.982·27-s − 0.752·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.492566119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.492566119\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 - 1.76T + 3T^{2} \) |
| 5 | \( 1 + 3.63T + 5T^{2} \) |
| 7 | \( 1 + 3.98T + 7T^{2} \) |
| 13 | \( 1 - 6.84T + 13T^{2} \) |
| 17 | \( 1 + 7.57T + 17T^{2} \) |
| 19 | \( 1 - 6.99T + 19T^{2} \) |
| 23 | \( 1 - 7.14T + 23T^{2} \) |
| 29 | \( 1 + 0.153T + 29T^{2} \) |
| 31 | \( 1 + 8.94T + 31T^{2} \) |
| 37 | \( 1 - 1.14T + 37T^{2} \) |
| 41 | \( 1 - 2.23T + 41T^{2} \) |
| 43 | \( 1 - 7.86T + 43T^{2} \) |
| 47 | \( 1 - 7.06T + 47T^{2} \) |
| 53 | \( 1 - 5.10T + 53T^{2} \) |
| 59 | \( 1 - 0.843T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 4.34T + 67T^{2} \) |
| 71 | \( 1 - 9.18T + 71T^{2} \) |
| 73 | \( 1 - 2.33T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 + 7.39T + 83T^{2} \) |
| 89 | \( 1 - 3.88T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−8.529239110235143591508122910913, −7.39324637797539807925381017433, −7.16141633774373681006542059228, −6.23556831380760860792537613187, −5.39783149521409702486787278567, −4.18156877270308553867102970275, −3.63190396195454197654194735089, −3.26765117996638077958815470679, −2.46990070520785492422237664167, −0.73986813030824870832416224983,
0.73986813030824870832416224983, 2.46990070520785492422237664167, 3.26765117996638077958815470679, 3.63190396195454197654194735089, 4.18156877270308553867102970275, 5.39783149521409702486787278567, 6.23556831380760860792537613187, 7.16141633774373681006542059228, 7.39324637797539807925381017433, 8.529239110235143591508122910913
