L(s) = 1 | + 13·7-s − 13-s − 11·19-s + 25·25-s + 46·31-s + 47·37-s + 22·43-s + 120·49-s − 121·61-s + 109·67-s − 97·73-s − 131·79-s − 13·91-s + 167·97-s + 37·103-s − 214·109-s + ⋯ |
L(s) = 1 | + 13/7·7-s − 0.0769·13-s − 0.578·19-s + 25-s + 1.48·31-s + 1.27·37-s + 0.511·43-s + 2.44·49-s − 1.98·61-s + 1.62·67-s − 1.32·73-s − 1.65·79-s − 1/7·91-s + 1.72·97-s + 0.359·103-s − 1.96·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.120417464\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.120417464\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( 1 - 13 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 + T + p^{2} T^{2} \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 + 11 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 - 46 T + p^{2} T^{2} \) |
| 37 | \( 1 - 47 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 - 22 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 + 121 T + p^{2} T^{2} \) |
| 67 | \( 1 - 109 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 97 T + p^{2} T^{2} \) |
| 79 | \( 1 + 131 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 - 167 T + p^{2} T^{2} \) |
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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
−11.03026690092992823481817781343, −10.21492171622462054868727733765, −8.933561478699626150595039599215, −8.199196890026437243064339513920, −7.42957089512152402146815923745, −6.17544996740251869284882079390, −4.97610347552061534285370424412, −4.30372489888907407074895129107, −2.55900693370604503446143988447, −1.23038637946345969613229333503,
1.23038637946345969613229333503, 2.55900693370604503446143988447, 4.30372489888907407074895129107, 4.97610347552061534285370424412, 6.17544996740251869284882079390, 7.42957089512152402146815923745, 8.199196890026437243064339513920, 8.933561478699626150595039599215, 10.21492171622462054868727733765, 11.03026690092992823481817781343