| L(s) = 1 | + 0.792·2-s + 3-s − 1.37·4-s + 3.37·5-s + 0.792·6-s + 2.52·7-s − 2.67·8-s + 9-s + 2.67·10-s − 1.37·12-s − 5.84·13-s + 2·14-s + 3.37·15-s + 0.627·16-s + 2.67·17-s + 0.792·18-s − 0.939·19-s − 4.62·20-s + 2.52·21-s + 2·23-s − 2.67·24-s + 6.37·25-s − 4.62·26-s + 27-s − 3.46·28-s − 0.792·29-s + 2.67·30-s + ⋯ |
| L(s) = 1 | + 0.560·2-s + 0.577·3-s − 0.686·4-s + 1.50·5-s + 0.323·6-s + 0.954·7-s − 0.944·8-s + 0.333·9-s + 0.844·10-s − 0.396·12-s − 1.61·13-s + 0.534·14-s + 0.870·15-s + 0.156·16-s + 0.648·17-s + 0.186·18-s − 0.215·19-s − 1.03·20-s + 0.550·21-s + 0.417·23-s − 0.545·24-s + 1.27·25-s − 0.907·26-s + 0.192·27-s − 0.654·28-s − 0.147·29-s + 0.487·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.253826192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.253826192\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.792T + 2T^{2} \) |
| 5 | \( 1 - 3.37T + 5T^{2} \) |
| 7 | \( 1 - 2.52T + 7T^{2} \) |
| 13 | \( 1 + 5.84T + 13T^{2} \) |
| 17 | \( 1 - 2.67T + 17T^{2} \) |
| 19 | \( 1 + 0.939T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 + 0.792T + 29T^{2} \) |
| 31 | \( 1 - 1.62T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 + 6.63T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 + 4.11T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 5.98T + 61T^{2} \) |
| 67 | \( 1 + 1.11T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 9.15T + 73T^{2} \) |
| 79 | \( 1 - 4.10T + 79T^{2} \) |
| 83 | \( 1 + 1.87T + 83T^{2} \) |
| 89 | \( 1 + 0.627T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−11.62050949706971693175784826730, −10.08291896927709444050110178215, −9.734244861019423555355439770966, −8.752974647539106053582913938070, −7.79694174501864191774960110430, −6.44663193842469751787444206692, −5.20396291391023309423150626190, −4.76468449858083751139902884752, −3.08416565161952874084754534841, −1.83411454730264611735408611709,
1.83411454730264611735408611709, 3.08416565161952874084754534841, 4.76468449858083751139902884752, 5.20396291391023309423150626190, 6.44663193842469751787444206692, 7.79694174501864191774960110430, 8.752974647539106053582913938070, 9.734244861019423555355439770966, 10.08291896927709444050110178215, 11.62050949706971693175784826730
