L(s) = 1 | + (0.178 − 0.178i)3-s + 0.542i·5-s + (−0.707 + 0.707i)7-s + 2.93i·9-s + (0.525 − 0.525i)11-s + (−2.31 + 2.31i)13-s + (0.0965 + 0.0965i)15-s + (0.596 + 0.596i)17-s + (−3.07 − 3.07i)19-s + 0.251i·21-s − 3.35·23-s + 4.70·25-s + (1.05 + 1.05i)27-s + (−6.56 + 6.56i)29-s − 8.61·31-s + ⋯ |
L(s) = 1 | + (0.102 − 0.102i)3-s + 0.242i·5-s + (−0.267 + 0.267i)7-s + 0.978i·9-s + (0.158 − 0.158i)11-s + (−0.641 + 0.641i)13-s + (0.0249 + 0.0249i)15-s + (0.144 + 0.144i)17-s + (−0.704 − 0.704i)19-s + 0.0549i·21-s − 0.700·23-s + 0.941·25-s + (0.203 + 0.203i)27-s + (−1.21 + 1.21i)29-s − 1.54·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9214039875\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9214039875\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
| 41 | \( 1 + (1.28 + 6.27i)T \) |
good | 3 | \( 1 + (-0.178 + 0.178i)T - 3iT^{2} \) |
| 5 | \( 1 - 0.542iT - 5T^{2} \) |
| 11 | \( 1 + (-0.525 + 0.525i)T - 11iT^{2} \) |
| 13 | \( 1 + (2.31 - 2.31i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.596 - 0.596i)T + 17iT^{2} \) |
| 19 | \( 1 + (3.07 + 3.07i)T + 19iT^{2} \) |
| 23 | \( 1 + 3.35T + 23T^{2} \) |
| 29 | \( 1 + (6.56 - 6.56i)T - 29iT^{2} \) |
| 31 | \( 1 + 8.61T + 31T^{2} \) |
| 37 | \( 1 + 1.51T + 37T^{2} \) |
| 43 | \( 1 - 7.10iT - 43T^{2} \) |
| 47 | \( 1 + (-4.92 - 4.92i)T + 47iT^{2} \) |
| 53 | \( 1 + (3.28 - 3.28i)T - 53iT^{2} \) |
| 59 | \( 1 - 9.28T + 59T^{2} \) |
| 61 | \( 1 - 6.51iT - 61T^{2} \) |
| 67 | \( 1 + (-1.81 - 1.81i)T + 67iT^{2} \) |
| 71 | \( 1 + (6.46 - 6.46i)T - 71iT^{2} \) |
| 73 | \( 1 - 12.1iT - 73T^{2} \) |
| 79 | \( 1 + (-10.2 + 10.2i)T - 79iT^{2} \) |
| 83 | \( 1 + 7.06T + 83T^{2} \) |
| 89 | \( 1 + (2.76 - 2.76i)T - 89iT^{2} \) |
| 97 | \( 1 + (7.05 + 7.05i)T + 97iT^{2} \) |
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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
−10.14122868491955436476229217905, −9.135788330490099756511715918348, −8.609913764902744672796927245811, −7.42871282338773875271037453688, −6.98017106681135029858087851178, −5.83319829201507258270602279222, −4.99833691634002301164373505481, −3.97101688580929132389976759123, −2.75067075658956440753738402278, −1.81361762274428093480179587612,
0.37323313909003264573017295699, 2.01280324460534345896068413803, 3.39716432313528319928871203936, 4.09277752372301411954316845140, 5.28217424586631049678654407534, 6.13832988180814054130476376356, 7.03722090106261790741922728191, 7.86691118299412701339979483976, 8.784946729510829815572734031564, 9.573964686232665386749031579050