| L(s) = 1 | − 2·2-s + 5.42·3-s + 4·4-s − 1.04·5-s − 10.8·6-s + 25.1·7-s − 8·8-s + 2.41·9-s + 2.09·10-s − 25.3·11-s + 21.6·12-s + 84.9·13-s − 50.2·14-s − 5.67·15-s + 16·16-s + 84.8·17-s − 4.83·18-s − 0.837·19-s − 4.18·20-s + 136.·21-s + 50.6·22-s + 62.2·23-s − 43.3·24-s − 123.·25-s − 169.·26-s − 133.·27-s + 100.·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.04·3-s + 0.5·4-s − 0.0936·5-s − 0.738·6-s + 1.35·7-s − 0.353·8-s + 0.0895·9-s + 0.0662·10-s − 0.693·11-s + 0.521·12-s + 1.81·13-s − 0.958·14-s − 0.0977·15-s + 0.250·16-s + 1.21·17-s − 0.0633·18-s − 0.0101·19-s − 0.0468·20-s + 1.41·21-s + 0.490·22-s + 0.563·23-s − 0.369·24-s − 0.991·25-s − 1.28·26-s − 0.950·27-s + 0.677·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.614437791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.614437791\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 37 | \( 1 + 37T \) |
| good | 3 | \( 1 - 5.42T + 27T^{2} \) |
| 5 | \( 1 + 1.04T + 125T^{2} \) |
| 7 | \( 1 - 25.1T + 343T^{2} \) |
| 11 | \( 1 + 25.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 84.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 84.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 0.837T + 6.85e3T^{2} \) |
| 23 | \( 1 - 62.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 167.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 191.T + 2.97e4T^{2} \) |
| 41 | \( 1 + 223.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 314.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 357.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 123.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 333.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 60.3T + 2.26e5T^{2} \) |
| 67 | \( 1 - 519.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 262.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 957.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 169.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 2.57T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.37e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.12e3T + 9.12e5T^{2} \) |
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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
−14.24742413200718565980114980634, −13.20192321946625548594624628122, −11.53792853550475425416381114684, −10.73398538248112268166131516710, −9.231115324466430228486916088747, −8.240512747768434046916387376937, −7.68307967775980104624470204821, −5.61424580293187930819073549423, −3.50179936631647083590577987510, −1.69156785543680390450576098167,
1.69156785543680390450576098167, 3.50179936631647083590577987510, 5.61424580293187930819073549423, 7.68307967775980104624470204821, 8.240512747768434046916387376937, 9.231115324466430228486916088747, 10.73398538248112268166131516710, 11.53792853550475425416381114684, 13.20192321946625548594624628122, 14.24742413200718565980114980634
