| L(s) = 1 | + 2-s + 0.348·3-s + 4-s − 5-s + 0.348·6-s + 4.26·7-s + 8-s − 2.87·9-s − 10-s − 11-s + 0.348·12-s + 3.68·13-s + 4.26·14-s − 0.348·15-s + 16-s + 4.99·17-s − 2.87·18-s + 2.58·19-s − 20-s + 1.48·21-s − 22-s − 0.990·23-s + 0.348·24-s + 25-s + 3.68·26-s − 2.04·27-s + 4.26·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.200·3-s + 0.5·4-s − 0.447·5-s + 0.142·6-s + 1.61·7-s + 0.353·8-s − 0.959·9-s − 0.316·10-s − 0.301·11-s + 0.100·12-s + 1.02·13-s + 1.14·14-s − 0.0898·15-s + 0.250·16-s + 1.21·17-s − 0.678·18-s + 0.594·19-s − 0.223·20-s + 0.324·21-s − 0.213·22-s − 0.206·23-s + 0.0710·24-s + 0.200·25-s + 0.722·26-s − 0.393·27-s + 0.806·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.248680550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.248680550\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 3 | \( 1 - 0.348T + 3T^{2} \) |
| 7 | \( 1 - 4.26T + 7T^{2} \) |
| 13 | \( 1 - 3.68T + 13T^{2} \) |
| 17 | \( 1 - 4.99T + 17T^{2} \) |
| 19 | \( 1 - 2.58T + 19T^{2} \) |
| 23 | \( 1 + 0.990T + 23T^{2} \) |
| 29 | \( 1 - 6.67T + 29T^{2} \) |
| 31 | \( 1 - 0.383T + 31T^{2} \) |
| 37 | \( 1 + 7.29T + 37T^{2} \) |
| 41 | \( 1 + 1.21T + 41T^{2} \) |
| 43 | \( 1 + 0.349T + 43T^{2} \) |
| 47 | \( 1 - 6.29T + 47T^{2} \) |
| 53 | \( 1 + 6.34T + 53T^{2} \) |
| 59 | \( 1 + 8.54T + 59T^{2} \) |
| 61 | \( 1 - 8.87T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 79 | \( 1 - 4.73T + 79T^{2} \) |
| 83 | \( 1 + 3.61T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 - 5.67T + 97T^{2} \) |
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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
−7.86830288882864692612640729964, −7.28095107843272835257263485678, −6.27874098417088088342319137020, −5.52599534619202071202791437073, −5.11204565847981622342697731560, −4.32344282312618368270027254019, −3.49367640551869763677685337926, −2.88871577866363266615140432623, −1.83236430440739391377391156066, −0.964330779100121407956614692445,
0.964330779100121407956614692445, 1.83236430440739391377391156066, 2.88871577866363266615140432623, 3.49367640551869763677685337926, 4.32344282312618368270027254019, 5.11204565847981622342697731560, 5.52599534619202071202791437073, 6.27874098417088088342319137020, 7.28095107843272835257263485678, 7.86830288882864692612640729964
