| L(s) = 1 | + 3-s + 3·5-s + 7-s + 9-s − 5·11-s + 3·15-s + 6·19-s + 21-s − 2·23-s + 4·25-s + 27-s − 9·29-s − 3·31-s − 5·33-s + 3·35-s + 6·37-s − 6·41-s − 4·43-s + 3·45-s − 6·47-s + 49-s − 3·53-s − 15·55-s + 6·57-s − 13·59-s + 4·61-s + 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 0.377·7-s + 1/3·9-s − 1.50·11-s + 0.774·15-s + 1.37·19-s + 0.218·21-s − 0.417·23-s + 4/5·25-s + 0.192·27-s − 1.67·29-s − 0.538·31-s − 0.870·33-s + 0.507·35-s + 0.986·37-s − 0.937·41-s − 0.609·43-s + 0.447·45-s − 0.875·47-s + 1/7·49-s − 0.412·53-s − 2.02·55-s + 0.794·57-s − 1.69·59-s + 0.512·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.758375629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.758375629\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−12.67458715050224, −12.05911266598191, −11.45913890763542, −11.04879972428221, −10.58420335564994, −10.09863146586811, −9.659454016453129, −9.440426544752047, −8.981686655427165, −8.202163976956219, −7.967205923109065, −7.526653334262020, −7.053540629011112, −6.384147589153245, −5.880041133908831, −5.460081131197432, −5.002687416231123, −4.744509658014375, −3.742987119645713, −3.353670290017975, −2.820919041056385, −2.078930429914491, −1.994583412198534, −1.292001451182954, −0.4467456310055771,
0.4467456310055771, 1.292001451182954, 1.994583412198534, 2.078930429914491, 2.820919041056385, 3.353670290017975, 3.742987119645713, 4.744509658014375, 5.002687416231123, 5.460081131197432, 5.880041133908831, 6.384147589153245, 7.053540629011112, 7.526653334262020, 7.967205923109065, 8.202163976956219, 8.981686655427165, 9.440426544752047, 9.659454016453129, 10.09863146586811, 10.58420335564994, 11.04879972428221, 11.45913890763542, 12.05911266598191, 12.67458715050224
