| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 14.9·5-s + 6·6-s − 1.95·7-s + 8·8-s + 9·9-s + 29.9·10-s − 0.954·11-s + 12·12-s + 48.8·13-s − 3.90·14-s + 44.8·15-s + 16·16-s + 12·17-s + 18·18-s + 59.8·20-s − 5.86·21-s − 1.90·22-s + 50.9·23-s + 24·24-s + 98.6·25-s + 97.6·26-s + 27·27-s − 7.81·28-s + 53.9·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.33·5-s + 0.408·6-s − 0.105·7-s + 0.353·8-s + 0.333·9-s + 0.945·10-s − 0.0261·11-s + 0.288·12-s + 1.04·13-s − 0.0746·14-s + 0.772·15-s + 0.250·16-s + 0.171·17-s + 0.235·18-s + 0.668·20-s − 0.0609·21-s − 0.0184·22-s + 0.461·23-s + 0.204·24-s + 0.789·25-s + 0.736·26-s + 0.192·27-s − 0.0527·28-s + 0.345·29-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
| \(L(2)\) |
\(\approx\) |
\(6.806283297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.806283297\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 14.9T + 125T^{2} \) |
| 7 | \( 1 + 1.95T + 343T^{2} \) |
| 11 | \( 1 + 0.954T + 1.33e3T^{2} \) |
| 13 | \( 1 - 48.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 12T + 4.91e3T^{2} \) |
| 23 | \( 1 - 50.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 53.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 11.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 176.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 253.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 488.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 351.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 184.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 280.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 563.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 188.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 106.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 982.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 804.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 873.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 122.T + 9.12e5T^{2} \) |
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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
−8.817403747501672572723392427748, −7.939890721503559199598722056738, −6.96427769531031205019921682204, −6.21470671795359006703315946054, −5.63720257380131919406757060082, −4.72855238333343793568290318702, −3.71565568552267502948551307659, −2.87060580589621042897933343527, −1.99881100713385024181758039819, −1.11844470731325284531868418061,
1.11844470731325284531868418061, 1.99881100713385024181758039819, 2.87060580589621042897933343527, 3.71565568552267502948551307659, 4.72855238333343793568290318702, 5.63720257380131919406757060082, 6.21470671795359006703315946054, 6.96427769531031205019921682204, 7.939890721503559199598722056738, 8.817403747501672572723392427748
