| L(s) = 1 | − 3·3-s + 18.5·5-s + 29.7·7-s + 9·9-s + 9.16·11-s + 80.3·13-s − 55.7·15-s − 31.1·17-s − 89.8·19-s − 89.2·21-s − 57.1·23-s + 220.·25-s − 27·27-s + 167.·29-s − 270.·31-s − 27.4·33-s + 552.·35-s − 157.·37-s − 240.·39-s − 404.·41-s + 317.·43-s + 167.·45-s + 63.1·47-s + 542.·49-s + 93.4·51-s + 616.·53-s + 170.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.66·5-s + 1.60·7-s + 0.333·9-s + 0.251·11-s + 1.71·13-s − 0.959·15-s − 0.444·17-s − 1.08·19-s − 0.927·21-s − 0.518·23-s + 1.76·25-s − 0.192·27-s + 1.06·29-s − 1.56·31-s − 0.145·33-s + 2.66·35-s − 0.699·37-s − 0.989·39-s − 1.53·41-s + 1.12·43-s + 0.554·45-s + 0.196·47-s + 1.58·49-s + 0.256·51-s + 1.59·53-s + 0.417·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.717549707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.717549707\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| good | 5 | \( 1 - 18.5T + 125T^{2} \) |
| 7 | \( 1 - 29.7T + 343T^{2} \) |
| 11 | \( 1 - 9.16T + 1.33e3T^{2} \) |
| 13 | \( 1 - 80.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 31.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 89.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 57.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 167.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 270.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 157.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 404.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 317.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 63.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 616.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 137.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 200.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 576.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 305.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 198.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 335.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 981.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 51.0T + 7.04e5T^{2} \) |
| 97 | \( 1 - 678.T + 9.12e5T^{2} \) |
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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
−10.80777615759458828133712862482, −10.30708944001699664719401642496, −8.947105425093209193288857141827, −8.403914390576521733610520968341, −6.86190294527252853913965042342, −5.96781827409578621957936629184, −5.27506804403011766720745152275, −4.12282741425448788101002408657, −2.06149860087989964408151722787, −1.31970139585693617995814790543,
1.31970139585693617995814790543, 2.06149860087989964408151722787, 4.12282741425448788101002408657, 5.27506804403011766720745152275, 5.96781827409578621957936629184, 6.86190294527252853913965042342, 8.403914390576521733610520968341, 8.947105425093209193288857141827, 10.30708944001699664719401642496, 10.80777615759458828133712862482
