L(s) = 1 | + (−0.729 − 1.21i)2-s + (1.74 − 0.724i)3-s + (−0.934 + 1.76i)4-s + (1.44 − 3.48i)5-s + (−2.15 − 1.59i)6-s + (0.707 + 0.707i)7-s + (2.82 − 0.158i)8-s + (0.414 − 0.414i)9-s + (−5.27 + 0.794i)10-s + (−2.63 − 1.09i)11-s + (−0.354 + 3.77i)12-s + (2.43 + 5.88i)13-s + (0.340 − 1.37i)14-s − 7.14i·15-s + (−2.25 − 3.30i)16-s − 6.50i·17-s + ⋯ |
L(s) = 1 | + (−0.516 − 0.856i)2-s + (1.01 − 0.418i)3-s + (−0.467 + 0.884i)4-s + (0.645 − 1.55i)5-s + (−0.879 − 0.649i)6-s + (0.267 + 0.267i)7-s + (0.998 − 0.0558i)8-s + (0.138 − 0.138i)9-s + (−1.66 + 0.251i)10-s + (−0.793 − 0.328i)11-s + (−0.102 + 1.08i)12-s + (0.675 + 1.63i)13-s + (0.0910 − 0.366i)14-s − 1.84i·15-s + (−0.563 − 0.826i)16-s − 1.57i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.828005 - 1.04861i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.828005 - 1.04861i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.729 + 1.21i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + (-1.74 + 0.724i)T + (2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (-1.44 + 3.48i)T + (-3.53 - 3.53i)T^{2} \) |
| 11 | \( 1 + (2.63 + 1.09i)T + (7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (-2.43 - 5.88i)T + (-9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + 6.50iT - 17T^{2} \) |
| 19 | \( 1 + (-0.581 - 1.40i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (3.10 - 3.10i)T - 23iT^{2} \) |
| 29 | \( 1 + (3.27 - 1.35i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 - 3.10T + 31T^{2} \) |
| 37 | \( 1 + (-0.403 + 0.973i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-5.07 + 5.07i)T - 41iT^{2} \) |
| 43 | \( 1 + (-4.48 - 1.85i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 - 9.40iT - 47T^{2} \) |
| 53 | \( 1 + (-3.60 - 1.49i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (3.33 - 8.04i)T + (-41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (-2.34 + 0.972i)T + (43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (0.931 - 0.385i)T + (47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (-7.12 - 7.12i)T + 71iT^{2} \) |
| 73 | \( 1 + (-3.38 + 3.38i)T - 73iT^{2} \) |
| 79 | \( 1 + 1.35iT - 79T^{2} \) |
| 83 | \( 1 + (1.64 + 3.96i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (8.99 + 8.99i)T + 89iT^{2} \) |
| 97 | \( 1 - 1.01T + 97T^{2} \) |
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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
−12.03609844004994373136150444102, −11.13868441494947039892059546891, −9.528246737958223029995786827614, −9.134450373371759772474725326219, −8.382650999600741062549546707650, −7.49240870228170073765845449699, −5.47352533190448181672612680028, −4.25935273566183382557850934936, −2.55317928123725888704357227897, −1.46706220249815093055100726702,
2.42690411936870669454549707779, 3.76653987366329402438654731739, 5.62515713278142412201596345592, 6.50280495286294592007553891405, 7.82138392026027666226965547400, 8.301614444851111640227623144633, 9.693084617958360211734424678054, 10.40316636500096959050962459707, 10.85198271605115026591054374013, 13.03888619794121314167845906355