| L(s) = 1 | − 3·3-s + 12.8·5-s + 24.8·7-s + 9·9-s + 19.6·11-s − 13·13-s − 38.4·15-s − 63.6·17-s − 0.832·19-s − 74.4·21-s + 119.·23-s + 39.6·25-s − 27·27-s − 6·29-s + 185.·31-s − 58.9·33-s + 318.·35-s + 143.·37-s + 39·39-s − 117.·41-s + 67.6·43-s + 115.·45-s + 476.·47-s + 273.·49-s + 190.·51-s + 59.3·53-s + 252.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.14·5-s + 1.34·7-s + 0.333·9-s + 0.539·11-s − 0.277·13-s − 0.662·15-s − 0.908·17-s − 0.0100·19-s − 0.774·21-s + 1.08·23-s + 0.317·25-s − 0.192·27-s − 0.0384·29-s + 1.07·31-s − 0.311·33-s + 1.53·35-s + 0.638·37-s + 0.160·39-s − 0.446·41-s + 0.240·43-s + 0.382·45-s + 1.47·47-s + 0.797·49-s + 0.524·51-s + 0.153·53-s + 0.618·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.244112721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.244112721\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 - 12.8T + 125T^{2} \) |
| 7 | \( 1 - 24.8T + 343T^{2} \) |
| 11 | \( 1 - 19.6T + 1.33e3T^{2} \) |
| 17 | \( 1 + 63.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 0.832T + 6.85e3T^{2} \) |
| 23 | \( 1 - 119.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 185.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 143.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 117.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 67.6T + 7.95e4T^{2} \) |
| 47 | \( 1 - 476.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 59.3T + 1.48e5T^{2} \) |
| 59 | \( 1 - 78T + 2.05e5T^{2} \) |
| 61 | \( 1 - 609.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 654.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 390.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 84.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 430.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.34e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 802.T + 9.12e5T^{2} \) |
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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
−11.22518561008651785748240627545, −10.43369018437925728476883753634, −9.417312769441868962911415642727, −8.514888941918950946430241952206, −7.23793385821445388787269117554, −6.22662863992314852118455930097, −5.24344062147135164953709995541, −4.38504921501161264833256947428, −2.33703125824516111387510416260, −1.18551823641953907802232410757,
1.18551823641953907802232410757, 2.33703125824516111387510416260, 4.38504921501161264833256947428, 5.24344062147135164953709995541, 6.22662863992314852118455930097, 7.23793385821445388787269117554, 8.514888941918950946430241952206, 9.417312769441868962911415642727, 10.43369018437925728476883753634, 11.22518561008651785748240627545
