L(s) = 1 | − 2.53·3-s + 1.47·5-s + 7-s + 3.43·9-s + 5.35·11-s − 3.55·13-s − 3.73·15-s − 2.73·17-s + 8.61·19-s − 2.53·21-s + 6.17·23-s − 2.83·25-s − 1.09·27-s + 8.49·29-s − 10.0·31-s − 13.5·33-s + 1.47·35-s − 0.333·37-s + 9.00·39-s − 1.30·41-s + 5.82·43-s + 5.04·45-s + 9.00·47-s + 49-s + 6.94·51-s + 1.20·53-s + 7.87·55-s + ⋯ |
L(s) = 1 | − 1.46·3-s + 0.658·5-s + 0.377·7-s + 1.14·9-s + 1.61·11-s − 0.985·13-s − 0.963·15-s − 0.664·17-s + 1.97·19-s − 0.553·21-s + 1.28·23-s − 0.567·25-s − 0.210·27-s + 1.57·29-s − 1.79·31-s − 2.36·33-s + 0.248·35-s − 0.0548·37-s + 1.44·39-s − 0.203·41-s + 0.887·43-s + 0.752·45-s + 1.31·47-s + 0.142·49-s + 0.972·51-s + 0.166·53-s + 1.06·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.475218393\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.475218393\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 + 2.53T + 3T^{2} \) |
| 5 | \( 1 - 1.47T + 5T^{2} \) |
| 11 | \( 1 - 5.35T + 11T^{2} \) |
| 13 | \( 1 + 3.55T + 13T^{2} \) |
| 17 | \( 1 + 2.73T + 17T^{2} \) |
| 19 | \( 1 - 8.61T + 19T^{2} \) |
| 23 | \( 1 - 6.17T + 23T^{2} \) |
| 29 | \( 1 - 8.49T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 0.333T + 37T^{2} \) |
| 41 | \( 1 + 1.30T + 41T^{2} \) |
| 43 | \( 1 - 5.82T + 43T^{2} \) |
| 47 | \( 1 - 9.00T + 47T^{2} \) |
| 53 | \( 1 - 1.20T + 53T^{2} \) |
| 59 | \( 1 + 2.71T + 59T^{2} \) |
| 61 | \( 1 + 4.48T + 61T^{2} \) |
| 67 | \( 1 + 8.45T + 67T^{2} \) |
| 71 | \( 1 - 2.46T + 71T^{2} \) |
| 73 | \( 1 + 6.31T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 5.41T + 83T^{2} \) |
| 89 | \( 1 - 6.44T + 89T^{2} \) |
| 97 | \( 1 - 4.95T + 97T^{2} \) |
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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
−8.843195731760825944534044038651, −7.34558220895261084303581672827, −7.09171386633886718121249915470, −6.17265368579765817332095464599, −5.58849633592980069381217928411, −4.93102782195218066803138071153, −4.22457961494760466367756335231, −2.98221322719782709029408396427, −1.66221930314456178350626692860, −0.825496096786516613207998089471,
0.825496096786516613207998089471, 1.66221930314456178350626692860, 2.98221322719782709029408396427, 4.22457961494760466367756335231, 4.93102782195218066803138071153, 5.58849633592980069381217928411, 6.17265368579765817332095464599, 7.09171386633886718121249915470, 7.34558220895261084303581672827, 8.843195731760825944534044038651