| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 3.36·5-s − 6·6-s + 26.7·7-s + 8·8-s + 9·9-s − 6.73·10-s + 5.26·11-s − 12·12-s + 66.1·13-s + 53.4·14-s + 10.1·15-s + 16·16-s + 23.2·17-s + 18·18-s + 19·19-s − 13.4·20-s − 80.2·21-s + 10.5·22-s − 2.63·23-s − 24·24-s − 113.·25-s + 132.·26-s − 27·27-s + 106.·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.301·5-s − 0.408·6-s + 1.44·7-s + 0.353·8-s + 0.333·9-s − 0.213·10-s + 0.144·11-s − 0.288·12-s + 1.41·13-s + 1.02·14-s + 0.173·15-s + 0.250·16-s + 0.331·17-s + 0.235·18-s + 0.229·19-s − 0.150·20-s − 0.833·21-s + 0.101·22-s − 0.0238·23-s − 0.204·24-s − 0.909·25-s + 0.997·26-s − 0.192·27-s + 0.721·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.291175804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.291175804\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 19 | \( 1 - 19T \) |
| good | 5 | \( 1 + 3.36T + 125T^{2} \) |
| 7 | \( 1 - 26.7T + 343T^{2} \) |
| 11 | \( 1 - 5.26T + 1.33e3T^{2} \) |
| 13 | \( 1 - 66.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 23.2T + 4.91e3T^{2} \) |
| 23 | \( 1 + 2.63T + 1.21e4T^{2} \) |
| 29 | \( 1 + 27.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 32.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 7.58T + 5.06e4T^{2} \) |
| 41 | \( 1 + 371.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 25.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 268.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 536.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 845.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 834.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 391.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 833.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 134.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 381.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 28.6T + 5.71e5T^{2} \) |
| 89 | \( 1 + 511.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.43e3T + 9.12e5T^{2} \) |
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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
−13.16699872900604414365544486631, −11.78736230409673550166071050141, −11.40757550701282967633316152726, −10.36065413255173942180600494237, −8.561874620253919370236687306301, −7.50478307297639675063864118723, −6.07809152070387833399153919842, −4.99562624930626439095107435857, −3.78954180722646165918180252330, −1.51985671706580427677858893552,
1.51985671706580427677858893552, 3.78954180722646165918180252330, 4.99562624930626439095107435857, 6.07809152070387833399153919842, 7.50478307297639675063864118723, 8.561874620253919370236687306301, 10.36065413255173942180600494237, 11.40757550701282967633316152726, 11.78736230409673550166071050141, 13.16699872900604414365544486631
