L(s) = 1 | − 2-s + 2.00·3-s + 4-s − 5-s − 2.00·6-s + 7-s − 8-s + 1.01·9-s + 10-s + 2.00·12-s + 4.02·13-s − 14-s − 2.00·15-s + 16-s − 0.300·17-s − 1.01·18-s − 7.73·19-s − 20-s + 2.00·21-s − 6.48·23-s − 2.00·24-s + 25-s − 4.02·26-s − 3.97·27-s + 28-s − 1.81·29-s + 2.00·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.15·3-s + 0.5·4-s − 0.447·5-s − 0.818·6-s + 0.377·7-s − 0.353·8-s + 0.339·9-s + 0.316·10-s + 0.578·12-s + 1.11·13-s − 0.267·14-s − 0.517·15-s + 0.250·16-s − 0.0728·17-s − 0.239·18-s − 1.77·19-s − 0.223·20-s + 0.437·21-s − 1.35·23-s − 0.409·24-s + 0.200·25-s − 0.790·26-s − 0.764·27-s + 0.188·28-s − 0.337·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 2.00T + 3T^{2} \) |
| 13 | \( 1 - 4.02T + 13T^{2} \) |
| 17 | \( 1 + 0.300T + 17T^{2} \) |
| 19 | \( 1 + 7.73T + 19T^{2} \) |
| 23 | \( 1 + 6.48T + 23T^{2} \) |
| 29 | \( 1 + 1.81T + 29T^{2} \) |
| 31 | \( 1 - 7.53T + 31T^{2} \) |
| 37 | \( 1 - 9.11T + 37T^{2} \) |
| 41 | \( 1 + 9.68T + 41T^{2} \) |
| 43 | \( 1 + 3.42T + 43T^{2} \) |
| 47 | \( 1 + 6.21T + 47T^{2} \) |
| 53 | \( 1 - 6.39T + 53T^{2} \) |
| 59 | \( 1 - 5.51T + 59T^{2} \) |
| 61 | \( 1 - 3.62T + 61T^{2} \) |
| 67 | \( 1 - 0.505T + 67T^{2} \) |
| 71 | \( 1 - 8.37T + 71T^{2} \) |
| 73 | \( 1 + 1.40T + 73T^{2} \) |
| 79 | \( 1 + 6.20T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 - 1.92T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−7.929197648090870461071986900963, −6.82733227781383901549810412023, −6.35121458077525786016015791999, −5.47741143046712229417022634395, −4.21852130259246353007590324593, −3.89994750658933222138591568328, −2.89317522561712016039164328629, −2.20329386492109827194972727999, −1.38870508801184222864486358109, 0,
1.38870508801184222864486358109, 2.20329386492109827194972727999, 2.89317522561712016039164328629, 3.89994750658933222138591568328, 4.21852130259246353007590324593, 5.47741143046712229417022634395, 6.35121458077525786016015791999, 6.82733227781383901549810412023, 7.929197648090870461071986900963