L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.84 − 3.20i)5-s − 0.999·8-s − 3.69·10-s + (−0.738 + 1.27i)11-s + (1.34 + 2.33i)13-s + (−0.5 + 0.866i)16-s − 6.57·17-s − 0.888·19-s + (−1.84 + 3.20i)20-s + (0.738 + 1.27i)22-s + (3.14 + 5.44i)23-s + (−4.34 + 7.52i)25-s + 2.69·26-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.827 − 1.43i)5-s − 0.353·8-s − 1.16·10-s + (−0.222 + 0.385i)11-s + (0.374 + 0.648i)13-s + (−0.125 + 0.216i)16-s − 1.59·17-s − 0.203·19-s + (−0.413 + 0.716i)20-s + (0.157 + 0.272i)22-s + (0.655 + 1.13i)23-s + (−0.868 + 1.50i)25-s + 0.529·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7670266553\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7670266553\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1.84 + 3.20i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.738 - 1.27i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.34 - 2.33i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 6.57T + 17T^{2} \) |
| 19 | \( 1 + 0.888T + 19T^{2} \) |
| 23 | \( 1 + (-3.14 - 5.44i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.25 - 2.17i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.40 - 5.89i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2.77T + 37T^{2} \) |
| 41 | \( 1 + (2.05 + 3.56i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.00618 + 0.0107i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.49 + 6.05i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 3.21T + 53T^{2} \) |
| 59 | \( 1 + (3.45 + 5.98i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.86 - 4.96i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.73 - 8.19i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 5.46T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 + (5.72 - 9.91i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.23 + 3.87i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 8.87T + 89T^{2} \) |
| 97 | \( 1 + (-6.58 + 11.4i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.860383961155017218395673946980, −8.511207432432065593885997976773, −7.43580897166223936412347302030, −6.65127783229098613886299621350, −5.49960295180897441354906766931, −4.76003152706565313714558018273, −4.27275974145317173924681500166, −3.42336300703873058217754732105, −2.08500063811174996808168102549, −1.11866734997559961697648318597,
0.24522628577142820703704552025, 2.48165559583459091655165003788, 3.10053816921755083762290948468, 4.07292421425174863511316567206, 4.72317835640381437460982402249, 6.07850103107823092022332007457, 6.41625589047958631050240470481, 7.21848790791921850721939890938, 7.913266596411233612462865585835, 8.488731853402373594114523073725