| L(s) = 1 | − 2-s + 1.16·3-s + 4-s − 1.76·5-s − 1.16·6-s − 8-s − 1.64·9-s + 1.76·10-s − 4.41·11-s + 1.16·12-s − 5.30·13-s − 2.05·15-s + 16-s − 0.211·17-s + 1.64·18-s + 5.92·19-s − 1.76·20-s + 4.41·22-s + 7.96·23-s − 1.16·24-s − 1.87·25-s + 5.30·26-s − 5.40·27-s − 29-s + 2.05·30-s − 4.52·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.671·3-s + 0.5·4-s − 0.790·5-s − 0.474·6-s − 0.353·8-s − 0.549·9-s + 0.558·10-s − 1.33·11-s + 0.335·12-s − 1.47·13-s − 0.530·15-s + 0.250·16-s − 0.0511·17-s + 0.388·18-s + 1.35·19-s − 0.395·20-s + 0.941·22-s + 1.66·23-s − 0.237·24-s − 0.375·25-s + 1.03·26-s − 1.03·27-s − 0.185·29-s + 0.374·30-s − 0.813·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8863388375\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8863388375\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 1.16T + 3T^{2} \) |
| 5 | \( 1 + 1.76T + 5T^{2} \) |
| 11 | \( 1 + 4.41T + 11T^{2} \) |
| 13 | \( 1 + 5.30T + 13T^{2} \) |
| 17 | \( 1 + 0.211T + 17T^{2} \) |
| 19 | \( 1 - 5.92T + 19T^{2} \) |
| 23 | \( 1 - 7.96T + 23T^{2} \) |
| 31 | \( 1 + 4.52T + 31T^{2} \) |
| 37 | \( 1 - 3.93T + 37T^{2} \) |
| 41 | \( 1 - 7.53T + 41T^{2} \) |
| 43 | \( 1 + 9.60T + 43T^{2} \) |
| 47 | \( 1 - 8.49T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 - 8.33T + 59T^{2} \) |
| 61 | \( 1 - 5.19T + 61T^{2} \) |
| 67 | \( 1 - 3.69T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 - 7.62T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 - 7.60T + 83T^{2} \) |
| 89 | \( 1 - 3.29T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−8.776808912007423609189366171073, −7.906954247864156883457912251249, −7.55224486687364352144409694494, −7.00765479337479825678741877543, −5.53481927538962070331257642432, −5.08141757691874179479580838381, −3.76016155336027590505807824341, −2.85194794283812208400962066810, −2.33447169333467886149645158058, −0.58954189480973097727406201329,
0.58954189480973097727406201329, 2.33447169333467886149645158058, 2.85194794283812208400962066810, 3.76016155336027590505807824341, 5.08141757691874179479580838381, 5.53481927538962070331257642432, 7.00765479337479825678741877543, 7.55224486687364352144409694494, 7.906954247864156883457912251249, 8.776808912007423609189366171073
