L(s) = 1 | + (0.939 − 0.342i)3-s + (0.766 − 0.642i)9-s + (0.939 + 1.62i)13-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.500 − 0.866i)27-s + (−0.173 + 0.300i)29-s + (0.173 + 0.300i)31-s + (1.43 + 1.20i)39-s + (−0.766 − 1.32i)41-s + (0.766 − 1.32i)47-s + (−0.5 − 0.866i)49-s + (−0.5 − 0.866i)59-s + (0.766 + 0.642i)69-s − 0.347·71-s + ⋯ |
L(s) = 1 | + (0.939 − 0.342i)3-s + (0.766 − 0.642i)9-s + (0.939 + 1.62i)13-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.500 − 0.866i)27-s + (−0.173 + 0.300i)29-s + (0.173 + 0.300i)31-s + (1.43 + 1.20i)39-s + (−0.766 − 1.32i)41-s + (0.766 − 1.32i)47-s + (−0.5 − 0.866i)49-s + (−0.5 − 0.866i)59-s + (0.766 + 0.642i)69-s − 0.347·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.869791826\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.869791826\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.939 + 0.342i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + 0.347T + T^{2} \) |
| 73 | \( 1 - 0.347T + T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−8.862011166630680930342202965799, −8.195318412665393463577684439726, −7.18322171449098925656216085381, −6.88243393139957331942768361825, −5.91784289283312042937136978267, −4.90284248535422175694123211522, −3.82967319322874241211766562271, −3.44376244381988360942480114083, −2.11564951866663963784259987469, −1.45426631760614867826947508116,
1.19595653589390158223348912010, 2.56058687177002190591109350044, 3.15132058807160964189710449331, 4.07840462782250195333438567221, 4.82913555189173550806650143482, 5.82793212095022638470309336204, 6.54118169006675636896898429275, 7.72142841579130517581123156505, 8.025162363964493541511551352562, 8.747628589069992566145807940433