| L(s) = 1 | − 1.70·2-s − 0.982·3-s + 0.895·4-s + 1.67·6-s + 5.03·7-s + 1.87·8-s − 2.03·9-s + 2.51·11-s − 0.880·12-s + 5.29·13-s − 8.56·14-s − 4.98·16-s − 5.08·17-s + 3.46·18-s + 3.09·19-s − 4.94·21-s − 4.28·22-s + 7.49·23-s − 1.84·24-s − 9.01·26-s + 4.94·27-s + 4.51·28-s + 8.95·29-s − 4.27·31-s + 4.73·32-s − 2.47·33-s + 8.65·34-s + ⋯ |
| L(s) = 1 | − 1.20·2-s − 0.567·3-s + 0.447·4-s + 0.682·6-s + 1.90·7-s + 0.664·8-s − 0.678·9-s + 0.759·11-s − 0.254·12-s + 1.46·13-s − 2.28·14-s − 1.24·16-s − 1.23·17-s + 0.816·18-s + 0.709·19-s − 1.07·21-s − 0.913·22-s + 1.56·23-s − 0.376·24-s − 1.76·26-s + 0.952·27-s + 0.852·28-s + 1.66·29-s − 0.767·31-s + 0.836·32-s − 0.430·33-s + 1.48·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.216563174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.216563174\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 197 | \( 1 - T \) |
| good | 2 | \( 1 + 1.70T + 2T^{2} \) |
| 3 | \( 1 + 0.982T + 3T^{2} \) |
| 7 | \( 1 - 5.03T + 7T^{2} \) |
| 11 | \( 1 - 2.51T + 11T^{2} \) |
| 13 | \( 1 - 5.29T + 13T^{2} \) |
| 17 | \( 1 + 5.08T + 17T^{2} \) |
| 19 | \( 1 - 3.09T + 19T^{2} \) |
| 23 | \( 1 - 7.49T + 23T^{2} \) |
| 29 | \( 1 - 8.95T + 29T^{2} \) |
| 31 | \( 1 + 4.27T + 31T^{2} \) |
| 37 | \( 1 - 0.741T + 37T^{2} \) |
| 41 | \( 1 + 1.33T + 41T^{2} \) |
| 43 | \( 1 - 9.49T + 43T^{2} \) |
| 47 | \( 1 + 2.68T + 47T^{2} \) |
| 53 | \( 1 - 1.33T + 53T^{2} \) |
| 59 | \( 1 - 0.910T + 59T^{2} \) |
| 61 | \( 1 + 5.51T + 61T^{2} \) |
| 67 | \( 1 - 2.79T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 + 6.18T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 - 3.68T + 89T^{2} \) |
| 97 | \( 1 - 8.00T + 97T^{2} \) |
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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
−8.514018134746900192816231540606, −7.78117234444899395448261220078, −6.94979725431502884073650500967, −6.26785582527633759606756079011, −5.23195128053892604489415183051, −4.74174326145237699721794320085, −3.89038229473508573528557701469, −2.48523006200615856326536295270, −1.36020398260320457395866743069, −0.899244788065390005946172557163,
0.899244788065390005946172557163, 1.36020398260320457395866743069, 2.48523006200615856326536295270, 3.89038229473508573528557701469, 4.74174326145237699721794320085, 5.23195128053892604489415183051, 6.26785582527633759606756079011, 6.94979725431502884073650500967, 7.78117234444899395448261220078, 8.514018134746900192816231540606
