| L(s) = 1 | + 2·2-s + 8.46·3-s + 4·4-s + 16.9·6-s + 8·8-s + 44.6·9-s + 5.60·11-s + 33.8·12-s + 44.1·13-s + 16·16-s − 109.·17-s + 89.2·18-s + 136.·19-s + 11.2·22-s − 21.4·23-s + 67.6·24-s + 88.3·26-s + 149.·27-s + 99.6·29-s + 17.1·31-s + 32·32-s + 47.4·33-s − 218.·34-s + 178.·36-s + 3.20·37-s + 273.·38-s + 373.·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.62·3-s + 0.5·4-s + 1.15·6-s + 0.353·8-s + 1.65·9-s + 0.153·11-s + 0.814·12-s + 0.942·13-s + 0.250·16-s − 1.56·17-s + 1.16·18-s + 1.65·19-s + 0.108·22-s − 0.194·23-s + 0.575·24-s + 0.666·26-s + 1.06·27-s + 0.638·29-s + 0.0992·31-s + 0.176·32-s + 0.250·33-s − 1.10·34-s + 0.826·36-s + 0.0142·37-s + 1.16·38-s + 1.53·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.890690102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.890690102\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 8.46T + 27T^{2} \) |
| 11 | \( 1 - 5.60T + 1.33e3T^{2} \) |
| 13 | \( 1 - 44.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 109.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 136.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 21.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 99.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 17.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 3.20T + 5.06e4T^{2} \) |
| 41 | \( 1 - 298.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 413.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 587.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 601.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 611.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 696.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 463.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 231.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 705.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.01e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 476.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 780.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 908.T + 9.12e5T^{2} \) |
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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
−8.430694804558294396250660863601, −7.988925260554496712651345529860, −7.05378711285441539621387297056, −6.42012457997125386114115132744, −5.31487933724269568693932488156, −4.30585194894654163211271048274, −3.68409776969679310685427451642, −2.89117584373050526796346509057, −2.14725545663316906599783916900, −1.09563197179743649761007669666,
1.09563197179743649761007669666, 2.14725545663316906599783916900, 2.89117584373050526796346509057, 3.68409776969679310685427451642, 4.30585194894654163211271048274, 5.31487933724269568693932488156, 6.42012457997125386114115132744, 7.05378711285441539621387297056, 7.988925260554496712651345529860, 8.430694804558294396250660863601
