| L(s) = 1 | + 0.507·2-s + 3-s − 1.74·4-s + 3.36·5-s + 0.507·6-s − 1.90·8-s + 9-s + 1.71·10-s + 3.52·11-s − 1.74·12-s − 0.582·13-s + 3.36·15-s + 2.51·16-s + 4.36·17-s + 0.507·18-s + 1.57·19-s − 5.86·20-s + 1.78·22-s − 23-s − 1.90·24-s + 6.35·25-s − 0.295·26-s + 27-s + 6.29·29-s + 1.71·30-s − 3.71·31-s + 5.07·32-s + ⋯ |
| L(s) = 1 | + 0.359·2-s + 0.577·3-s − 0.871·4-s + 1.50·5-s + 0.207·6-s − 0.671·8-s + 0.333·9-s + 0.541·10-s + 1.06·11-s − 0.502·12-s − 0.161·13-s + 0.869·15-s + 0.629·16-s + 1.05·17-s + 0.119·18-s + 0.361·19-s − 1.31·20-s + 0.381·22-s − 0.208·23-s − 0.387·24-s + 1.27·25-s − 0.0579·26-s + 0.192·27-s + 1.16·29-s + 0.312·30-s − 0.667·31-s + 0.898·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.326671311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.326671311\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 0.507T + 2T^{2} \) |
| 5 | \( 1 - 3.36T + 5T^{2} \) |
| 11 | \( 1 - 3.52T + 11T^{2} \) |
| 13 | \( 1 + 0.582T + 13T^{2} \) |
| 17 | \( 1 - 4.36T + 17T^{2} \) |
| 19 | \( 1 - 1.57T + 19T^{2} \) |
| 29 | \( 1 - 6.29T + 29T^{2} \) |
| 31 | \( 1 + 3.71T + 31T^{2} \) |
| 37 | \( 1 + 5.02T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 3.16T + 43T^{2} \) |
| 47 | \( 1 + 12.5T + 47T^{2} \) |
| 53 | \( 1 + 5.42T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 9.68T + 67T^{2} \) |
| 71 | \( 1 - 8.15T + 71T^{2} \) |
| 73 | \( 1 - 3.05T + 73T^{2} \) |
| 79 | \( 1 + 5.16T + 79T^{2} \) |
| 83 | \( 1 + 0.437T + 83T^{2} \) |
| 89 | \( 1 - 8.36T + 89T^{2} \) |
| 97 | \( 1 - 6.27T + 97T^{2} \) |
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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
−8.662546598108303714202517704159, −8.159065249212780923892908209858, −6.91475323079297340694243876032, −6.32780335299655064695731766662, −5.39208390115235353808789377647, −4.97593067566906883171044845211, −3.80059793510247034359006353025, −3.19855448048612052032354715039, −2.02830335015444566955575901363, −1.10036700646671570764338202068,
1.10036700646671570764338202068, 2.02830335015444566955575901363, 3.19855448048612052032354715039, 3.80059793510247034359006353025, 4.97593067566906883171044845211, 5.39208390115235353808789377647, 6.32780335299655064695731766662, 6.91475323079297340694243876032, 8.159065249212780923892908209858, 8.662546598108303714202517704159
