L(s) = 1 | − 2-s + 2.95·3-s + 4-s + 2.26·5-s − 2.95·6-s − 8-s + 5.70·9-s − 2.26·10-s − 3.09·11-s + 2.95·12-s + 0.322·13-s + 6.68·15-s + 16-s + 2.44·17-s − 5.70·18-s + 4.20·19-s + 2.26·20-s + 3.09·22-s + 8.85·23-s − 2.95·24-s + 0.128·25-s − 0.322·26-s + 7.99·27-s + 1.00·29-s − 6.68·30-s + 5.72·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.70·3-s + 0.5·4-s + 1.01·5-s − 1.20·6-s − 0.353·8-s + 1.90·9-s − 0.716·10-s − 0.932·11-s + 0.851·12-s + 0.0893·13-s + 1.72·15-s + 0.250·16-s + 0.593·17-s − 1.34·18-s + 0.964·19-s + 0.506·20-s + 0.659·22-s + 1.84·23-s − 0.602·24-s + 0.0256·25-s − 0.0632·26-s + 1.53·27-s + 0.187·29-s − 1.22·30-s + 1.02·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.376850522\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.376850522\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 - 2.95T + 3T^{2} \) |
| 5 | \( 1 - 2.26T + 5T^{2} \) |
| 11 | \( 1 + 3.09T + 11T^{2} \) |
| 13 | \( 1 - 0.322T + 13T^{2} \) |
| 17 | \( 1 - 2.44T + 17T^{2} \) |
| 19 | \( 1 - 4.20T + 19T^{2} \) |
| 23 | \( 1 - 8.85T + 23T^{2} \) |
| 29 | \( 1 - 1.00T + 29T^{2} \) |
| 31 | \( 1 - 5.72T + 31T^{2} \) |
| 37 | \( 1 + 6.80T + 37T^{2} \) |
| 43 | \( 1 - 0.700T + 43T^{2} \) |
| 47 | \( 1 + 0.0829T + 47T^{2} \) |
| 53 | \( 1 + 6.32T + 53T^{2} \) |
| 59 | \( 1 + 9.98T + 59T^{2} \) |
| 61 | \( 1 - 2.40T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 - 8.34T + 73T^{2} \) |
| 79 | \( 1 - 2.11T + 79T^{2} \) |
| 83 | \( 1 + 0.591T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 - 4.04T + 97T^{2} \) |
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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
−8.604119762409480026576531406211, −7.74753226788222632467817026356, −7.40725583869427183004295222964, −6.47213240945216210593692233926, −5.49625868124313132192652168090, −4.67808119128558903526542614642, −3.22450505669254195218854349210, −2.95399799938444344273798013406, −2.01517167185440915650878801118, −1.17678287563820151936338581690,
1.17678287563820151936338581690, 2.01517167185440915650878801118, 2.95399799938444344273798013406, 3.22450505669254195218854349210, 4.67808119128558903526542614642, 5.49625868124313132192652168090, 6.47213240945216210593692233926, 7.40725583869427183004295222964, 7.74753226788222632467817026356, 8.604119762409480026576531406211