L(s) = 1 | − 2-s + 2.89·3-s + 4-s + 4.15·5-s − 2.89·6-s − 4.06·7-s − 8-s + 5.40·9-s − 4.15·10-s + 1.29·11-s + 2.89·12-s + 1.51·13-s + 4.06·14-s + 12.0·15-s + 16-s + 2.32·17-s − 5.40·18-s − 4.02·19-s + 4.15·20-s − 11.7·21-s − 1.29·22-s − 2.94·23-s − 2.89·24-s + 12.2·25-s − 1.51·26-s + 6.97·27-s − 4.06·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.67·3-s + 0.5·4-s + 1.85·5-s − 1.18·6-s − 1.53·7-s − 0.353·8-s + 1.80·9-s − 1.31·10-s + 0.389·11-s + 0.836·12-s + 0.420·13-s + 1.08·14-s + 3.11·15-s + 0.250·16-s + 0.564·17-s − 1.27·18-s − 0.922·19-s + 0.929·20-s − 2.57·21-s − 0.275·22-s − 0.614·23-s − 0.591·24-s + 2.45·25-s − 0.297·26-s + 1.34·27-s − 0.768·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.408371787\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.408371787\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
good | 3 | \( 1 - 2.89T + 3T^{2} \) |
| 5 | \( 1 - 4.15T + 5T^{2} \) |
| 7 | \( 1 + 4.06T + 7T^{2} \) |
| 11 | \( 1 - 1.29T + 11T^{2} \) |
| 13 | \( 1 - 1.51T + 13T^{2} \) |
| 17 | \( 1 - 2.32T + 17T^{2} \) |
| 19 | \( 1 + 4.02T + 19T^{2} \) |
| 23 | \( 1 + 2.94T + 23T^{2} \) |
| 29 | \( 1 - 7.18T + 29T^{2} \) |
| 31 | \( 1 - 8.26T + 31T^{2} \) |
| 37 | \( 1 + 1.70T + 37T^{2} \) |
| 41 | \( 1 - 7.80T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 8.57T + 47T^{2} \) |
| 53 | \( 1 - 6.46T + 53T^{2} \) |
| 59 | \( 1 + 7.33T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 5.67T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 + 4.79T + 79T^{2} \) |
| 83 | \( 1 - 4.17T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−8.785938294699379413765217219575, −7.968164476526359407478671574334, −6.95501450439448040274540670470, −6.32325488758186976558635861396, −5.95834989700500633264147060868, −4.48595015890048494705068206292, −3.34151362491517064897274204289, −2.78207438688865716761149616649, −2.11602275984007188201997783333, −1.16252730926926180395942766878,
1.16252730926926180395942766878, 2.11602275984007188201997783333, 2.78207438688865716761149616649, 3.34151362491517064897274204289, 4.48595015890048494705068206292, 5.95834989700500633264147060868, 6.32325488758186976558635861396, 6.95501450439448040274540670470, 7.968164476526359407478671574334, 8.785938294699379413765217219575