| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 4.18·5-s − 6·6-s + 3.18·7-s − 8·8-s + 9·9-s − 8.37·10-s + 69.4·11-s + 12·12-s + 8.12·13-s − 6.37·14-s + 12.5·15-s + 16·16-s − 106.·17-s − 18·18-s + 16.7·20-s + 9.56·21-s − 138.·22-s − 176.·23-s − 24·24-s − 107.·25-s − 16.2·26-s + 27·27-s + 12.7·28-s + 66.2·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.374·5-s − 0.408·6-s + 0.172·7-s − 0.353·8-s + 0.333·9-s − 0.264·10-s + 1.90·11-s + 0.288·12-s + 0.173·13-s − 0.121·14-s + 0.216·15-s + 0.250·16-s − 1.51·17-s − 0.235·18-s + 0.187·20-s + 0.0993·21-s − 1.34·22-s − 1.60·23-s − 0.204·24-s − 0.859·25-s − 0.122·26-s + 0.192·27-s + 0.0860·28-s + 0.424·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.482779116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.482779116\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 4.18T + 125T^{2} \) |
| 7 | \( 1 - 3.18T + 343T^{2} \) |
| 11 | \( 1 - 69.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 8.12T + 2.19e3T^{2} \) |
| 17 | \( 1 + 106.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 176.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 66.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 140.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 156.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 414.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 115.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 620.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 371.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 91.6T + 2.05e5T^{2} \) |
| 61 | \( 1 - 218.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 145.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 887.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 199.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 389.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 380.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 425.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 419.T + 9.12e5T^{2} \) |
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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
−8.867943302029001833553618050762, −8.111086174599574812646693988496, −7.27385059292175093766341880848, −6.38642282094283099263239180693, −5.98314385669957276898892914281, −4.34693911501265531622379814701, −3.91495861166455511854737824621, −2.50910391555321575753976524915, −1.81668553488916473749729846344, −0.791482426939584380428333117250,
0.791482426939584380428333117250, 1.81668553488916473749729846344, 2.50910391555321575753976524915, 3.91495861166455511854737824621, 4.34693911501265531622379814701, 5.98314385669957276898892914281, 6.38642282094283099263239180693, 7.27385059292175093766341880848, 8.111086174599574812646693988496, 8.867943302029001833553618050762
