| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 17.4·5-s + 6·6-s − 0.298·7-s + 8·8-s + 9·9-s + 34.8·10-s + 53.7·11-s + 12·12-s + 45.9·13-s − 0.596·14-s + 52.2·15-s + 16·16-s + 87.9·17-s + 18·18-s + 69.6·20-s − 0.894·21-s + 107.·22-s − 65.1·23-s + 24·24-s + 178.·25-s + 91.8·26-s + 27·27-s − 1.19·28-s + 37.2·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.55·5-s + 0.408·6-s − 0.0160·7-s + 0.353·8-s + 0.333·9-s + 1.10·10-s + 1.47·11-s + 0.288·12-s + 0.980·13-s − 0.0113·14-s + 0.899·15-s + 0.250·16-s + 1.25·17-s + 0.235·18-s + 0.778·20-s − 0.00929·21-s + 1.04·22-s − 0.590·23-s + 0.204·24-s + 1.42·25-s + 0.692·26-s + 0.192·27-s − 0.00804·28-s + 0.238·29-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
| \(L(2)\) |
\(\approx\) |
\(7.869164405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.869164405\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 17.4T + 125T^{2} \) |
| 7 | \( 1 + 0.298T + 343T^{2} \) |
| 11 | \( 1 - 53.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 45.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 87.9T + 4.91e3T^{2} \) |
| 23 | \( 1 + 65.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 37.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 176.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 153.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 171.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 248.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 372.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 354.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 734.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 172.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 277.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 868.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 844.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 333.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.41e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.32e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.47e3T + 9.12e5T^{2} \) |
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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
−8.901398693543904304240467285560, −7.929824974648069350217297911006, −6.92610143333856336348447788124, −6.12252063627846117250416901314, −5.79320487510126736530497757240, −4.66868339163344841861336642580, −3.69020929467408513174988980569, −2.99116506430341042573524668525, −1.72788618263764140206522344480, −1.34829235522092014266610991433,
1.34829235522092014266610991433, 1.72788618263764140206522344480, 2.99116506430341042573524668525, 3.69020929467408513174988980569, 4.66868339163344841861336642580, 5.79320487510126736530497757240, 6.12252063627846117250416901314, 6.92610143333856336348447788124, 7.929824974648069350217297911006, 8.901398693543904304240467285560
