| L(s) = 1 | − 2-s − 1.16·3-s + 4-s − 0.683·5-s + 1.16·6-s − 0.316·7-s − 8-s − 1.64·9-s + 0.683·10-s + 11-s − 1.16·12-s + 0.316·13-s + 0.316·14-s + 0.796·15-s + 16-s − 1.84·17-s + 1.64·18-s − 0.683·20-s + 0.367·21-s − 22-s − 4.48·23-s + 1.16·24-s − 4.53·25-s − 0.316·26-s + 5.40·27-s − 0.316·28-s − 5.49·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.672·3-s + 0.5·4-s − 0.305·5-s + 0.475·6-s − 0.119·7-s − 0.353·8-s − 0.548·9-s + 0.216·10-s + 0.301·11-s − 0.336·12-s + 0.0876·13-s + 0.0844·14-s + 0.205·15-s + 0.250·16-s − 0.448·17-s + 0.387·18-s − 0.152·20-s + 0.0802·21-s − 0.213·22-s − 0.934·23-s + 0.237·24-s − 0.906·25-s − 0.0619·26-s + 1.04·27-s − 0.0597·28-s − 1.01·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4400589616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4400589616\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 1.16T + 3T^{2} \) |
| 5 | \( 1 + 0.683T + 5T^{2} \) |
| 7 | \( 1 + 0.316T + 7T^{2} \) |
| 13 | \( 1 - 0.316T + 13T^{2} \) |
| 17 | \( 1 + 1.84T + 17T^{2} \) |
| 23 | \( 1 + 4.48T + 23T^{2} \) |
| 29 | \( 1 + 5.49T + 29T^{2} \) |
| 31 | \( 1 + 0.796T + 31T^{2} \) |
| 37 | \( 1 + 0.632T + 37T^{2} \) |
| 41 | \( 1 - 3.83T + 41T^{2} \) |
| 43 | \( 1 - 6.68T + 43T^{2} \) |
| 47 | \( 1 + 4.32T + 47T^{2} \) |
| 53 | \( 1 - 3.44T + 53T^{2} \) |
| 59 | \( 1 - 2.80T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 3.44T + 73T^{2} \) |
| 79 | \( 1 + 9.06T + 79T^{2} \) |
| 83 | \( 1 + 4.35T + 83T^{2} \) |
| 89 | \( 1 - 9.28T + 89T^{2} \) |
| 97 | \( 1 - 3.69T + 97T^{2} \) |
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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
−7.81775204581138292433829291465, −7.28077589003246979403689859411, −6.31186176379087708586335139108, −6.02202630097050379836892775554, −5.21049197326161583910976847411, −4.25942378600726445877104908585, −3.50906568857825699452902708015, −2.51896623093295714311668866993, −1.60758992607344691037154983004, −0.37554649549721876270480340782,
0.37554649549721876270480340782, 1.60758992607344691037154983004, 2.51896623093295714311668866993, 3.50906568857825699452902708015, 4.25942378600726445877104908585, 5.21049197326161583910976847411, 6.02202630097050379836892775554, 6.31186176379087708586335139108, 7.28077589003246979403689859411, 7.81775204581138292433829291465
