| L(s) = 1 | − 1.71·2-s − 1.27·3-s + 0.944·4-s − 2.00·5-s + 2.19·6-s − 3.76·7-s + 1.81·8-s − 1.36·9-s + 3.44·10-s − 2.15·11-s − 1.20·12-s + 13-s + 6.46·14-s + 2.56·15-s − 4.99·16-s + 0.496·17-s + 2.34·18-s − 2.24·19-s − 1.89·20-s + 4.81·21-s + 3.69·22-s + 5.01·23-s − 2.31·24-s − 0.966·25-s − 1.71·26-s + 5.58·27-s − 3.55·28-s + ⋯ |
| L(s) = 1 | − 1.21·2-s − 0.738·3-s + 0.472·4-s − 0.898·5-s + 0.895·6-s − 1.42·7-s + 0.640·8-s − 0.455·9-s + 1.08·10-s − 0.649·11-s − 0.348·12-s + 0.277·13-s + 1.72·14-s + 0.662·15-s − 1.24·16-s + 0.120·17-s + 0.552·18-s − 0.514·19-s − 0.424·20-s + 1.05·21-s + 0.787·22-s + 1.04·23-s − 0.472·24-s − 0.193·25-s − 0.336·26-s + 1.07·27-s − 0.672·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2165804613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2165804613\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.71T + 2T^{2} \) |
| 3 | \( 1 + 1.27T + 3T^{2} \) |
| 5 | \( 1 + 2.00T + 5T^{2} \) |
| 7 | \( 1 + 3.76T + 7T^{2} \) |
| 11 | \( 1 + 2.15T + 11T^{2} \) |
| 17 | \( 1 - 0.496T + 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 23 | \( 1 - 5.01T + 23T^{2} \) |
| 29 | \( 1 - 2.58T + 29T^{2} \) |
| 37 | \( 1 - 0.294T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 1.81T + 43T^{2} \) |
| 47 | \( 1 + 0.507T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 + 9.27T + 59T^{2} \) |
| 61 | \( 1 - 0.950T + 61T^{2} \) |
| 67 | \( 1 + 3.55T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 + 0.201T + 73T^{2} \) |
| 79 | \( 1 + 8.22T + 79T^{2} \) |
| 83 | \( 1 - 8.35T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−10.94598555589101269205531670401, −10.43548813970371528362754300778, −9.389542288195352897955976707300, −8.623798583466274128938081095251, −7.65815306263460742791287368204, −6.76996979860847400647079482248, −5.72080172977349753528120057573, −4.32476162126708797611484532789, −2.91905830421569518123219277428, −0.52620303470163532903561706855,
0.52620303470163532903561706855, 2.91905830421569518123219277428, 4.32476162126708797611484532789, 5.72080172977349753528120057573, 6.76996979860847400647079482248, 7.65815306263460742791287368204, 8.623798583466274128938081095251, 9.389542288195352897955976707300, 10.43548813970371528362754300778, 10.94598555589101269205531670401
