L(s) = 1 | + 2.51·2-s − 2.82·3-s + 4.33·4-s + 2.87·5-s − 7.11·6-s − 3.18·7-s + 5.87·8-s + 5.00·9-s + 7.23·10-s + 1.31·11-s − 12.2·12-s + 6.51·13-s − 8.01·14-s − 8.13·15-s + 6.11·16-s + 0.932·17-s + 12.5·18-s + 0.348·19-s + 12.4·20-s + 9.01·21-s + 3.31·22-s + 23-s − 16.6·24-s + 3.27·25-s + 16.3·26-s − 5.65·27-s − 13.8·28-s + ⋯ |
L(s) = 1 | + 1.77·2-s − 1.63·3-s + 2.16·4-s + 1.28·5-s − 2.90·6-s − 1.20·7-s + 2.07·8-s + 1.66·9-s + 2.28·10-s + 0.397·11-s − 3.53·12-s + 1.80·13-s − 2.14·14-s − 2.10·15-s + 1.52·16-s + 0.226·17-s + 2.96·18-s + 0.0799·19-s + 2.78·20-s + 1.96·21-s + 0.706·22-s + 0.208·23-s − 3.39·24-s + 0.654·25-s + 3.21·26-s − 1.08·27-s − 2.60·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.040570571\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.040570571\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 - 2.51T + 2T^{2} \) |
| 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 - 2.87T + 5T^{2} \) |
| 7 | \( 1 + 3.18T + 7T^{2} \) |
| 11 | \( 1 - 1.31T + 11T^{2} \) |
| 13 | \( 1 - 6.51T + 13T^{2} \) |
| 17 | \( 1 - 0.932T + 17T^{2} \) |
| 19 | \( 1 - 0.348T + 19T^{2} \) |
| 31 | \( 1 - 4.04T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 2.12T + 41T^{2} \) |
| 43 | \( 1 + 3.91T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 6.92T + 53T^{2} \) |
| 59 | \( 1 + 7.17T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 - 3.95T + 67T^{2} \) |
| 71 | \( 1 + 15.7T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 4.56T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−10.69775959949444998195346782013, −10.22629967198852389250389998680, −9.029803748666920176453522487877, −6.95530717997591134553088409193, −6.38249808760685493965018127923, −5.89600990654803267790763583289, −5.37960844889860034061390077753, −4.18089945197384520468085245327, −3.15523752818061099119049601502, −1.49717067104105362621042922726,
1.49717067104105362621042922726, 3.15523752818061099119049601502, 4.18089945197384520468085245327, 5.37960844889860034061390077753, 5.89600990654803267790763583289, 6.38249808760685493965018127923, 6.95530717997591134553088409193, 9.029803748666920176453522487877, 10.22629967198852389250389998680, 10.69775959949444998195346782013