| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 0.909·5-s + 6·6-s − 16.3·7-s − 8·8-s + 9·9-s + 1.81·10-s − 1.94·11-s − 12·12-s − 15.0·13-s + 32.6·14-s + 2.72·15-s + 16·16-s + 32.0·17-s − 18·18-s − 3.63·20-s + 48.9·21-s + 3.88·22-s − 44.8·23-s + 24·24-s − 124.·25-s + 30.1·26-s − 27·27-s − 65.3·28-s − 31.1·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.0813·5-s + 0.408·6-s − 0.881·7-s − 0.353·8-s + 0.333·9-s + 0.0575·10-s − 0.0532·11-s − 0.288·12-s − 0.321·13-s + 0.623·14-s + 0.0469·15-s + 0.250·16-s + 0.456·17-s − 0.235·18-s − 0.0406·20-s + 0.508·21-s + 0.0376·22-s − 0.406·23-s + 0.204·24-s − 0.993·25-s + 0.227·26-s − 0.192·27-s − 0.440·28-s − 0.199·29-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 0.909T + 125T^{2} \) |
| 7 | \( 1 + 16.3T + 343T^{2} \) |
| 11 | \( 1 + 1.94T + 1.33e3T^{2} \) |
| 13 | \( 1 + 15.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 32.0T + 4.91e3T^{2} \) |
| 23 | \( 1 + 44.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 31.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 299.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 252.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 81.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 217.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 187.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 11.8T + 1.48e5T^{2} \) |
| 59 | \( 1 + 244.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 526.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 210.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 503.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 115.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 220.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 200.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 174.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.10e3T + 9.12e5T^{2} \) |
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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
−8.233637849477821968614910306664, −7.65240212465732974196272479216, −6.69390117222560602127687952688, −6.19009496234630594361169576298, −5.32611046629390077947355774805, −4.23278744789764541215029951358, −3.23134069995633458518212389062, −2.21440901149163738246885046366, −0.939793968710059007423980471979, 0,
0.939793968710059007423980471979, 2.21440901149163738246885046366, 3.23134069995633458518212389062, 4.23278744789764541215029951358, 5.32611046629390077947355774805, 6.19009496234630594361169576298, 6.69390117222560602127687952688, 7.65240212465732974196272479216, 8.233637849477821968614910306664
