L(s) = 1 | + (−0.767 + 1.18i)2-s + (−0.822 − 1.82i)4-s + 0.841i·5-s + 2.64·7-s + (2.79 + 0.420i)8-s + (−0.999 − 0.645i)10-s + 3.91i·11-s + 0.645i·13-s + (−2.02 + 3.14i)14-s + (−2.64 + 2.99i)16-s + 5.59·17-s + 4.29i·19-s + (1.53 − 0.692i)20-s + (−4.64 − 3.00i)22-s − 8.66·23-s + ⋯ |
L(s) = 1 | + (−0.542 + 0.840i)2-s + (−0.411 − 0.911i)4-s + 0.376i·5-s + 0.999·7-s + (0.988 + 0.148i)8-s + (−0.316 − 0.204i)10-s + 1.17i·11-s + 0.179i·13-s + (−0.542 + 0.840i)14-s + (−0.661 + 0.749i)16-s + 1.35·17-s + 0.984i·19-s + (0.343 − 0.154i)20-s + (−0.990 − 0.639i)22-s − 1.80·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.744887 + 0.641187i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.744887 + 0.641187i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.767 - 1.18i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 0.841iT - 5T^{2} \) |
| 7 | \( 1 - 2.64T + 7T^{2} \) |
| 11 | \( 1 - 3.91iT - 11T^{2} \) |
| 13 | \( 1 - 0.645iT - 13T^{2} \) |
| 17 | \( 1 - 5.59T + 17T^{2} \) |
| 19 | \( 1 - 4.29iT - 19T^{2} \) |
| 23 | \( 1 + 8.66T + 23T^{2} \) |
| 29 | \( 1 + 7.82iT - 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 6.64iT - 37T^{2} \) |
| 41 | \( 1 - 6.13T + 41T^{2} \) |
| 43 | \( 1 + 7.29iT - 43T^{2} \) |
| 47 | \( 1 + 2.52T + 47T^{2} \) |
| 53 | \( 1 + 7.82iT - 53T^{2} \) |
| 59 | \( 1 + 7.27iT - 59T^{2} \) |
| 61 | \( 1 + 13.9iT - 61T^{2} \) |
| 67 | \( 1 - 3iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 9.58T + 73T^{2} \) |
| 79 | \( 1 + 3.35T + 79T^{2} \) |
| 83 | \( 1 - 9.50iT - 83T^{2} \) |
| 89 | \( 1 + 5.59T + 89T^{2} \) |
| 97 | \( 1 + 8.29T + 97T^{2} \) |
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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
−12.45080661991274548631935903809, −11.49293994175779057089015434498, −10.19015783691285162158386780764, −9.769919048500934709208584538178, −8.175796898796532629910597039145, −7.75497577641718738890684829853, −6.53080121023081450317818940402, −5.38521962201029377332191432369, −4.22408509057370578509441974476, −1.80427050584664268243234755352,
1.19433220821260657188266070116, 2.97405247052653285110527425518, 4.39653136774595535225486369777, 5.64778897199645270470709364906, 7.49416147952652700148558789143, 8.329228002140325844678845116028, 9.089845683171426466334342541915, 10.33477334929508851176797941341, 11.11145571443820875674856311441, 11.93303970696509773633134903141