| L(s) = 1 | + (3.37 + 1.94i)5-s + (6.97 − 0.619i)7-s + (−9.62 + 5.55i)11-s − 17.2·13-s + (−19.3 + 11.1i)17-s + (−8.62 + 14.9i)19-s + (−22.6 − 13.1i)23-s + (−4.89 − 8.47i)25-s + 8.73i·29-s + (−22.0 − 38.1i)31-s + (24.7 + 11.5i)35-s + (−19.3 + 33.4i)37-s + 44.1i·41-s + 31.1·43-s + (−12.8 − 7.43i)47-s + ⋯ |
| L(s) = 1 | + (0.675 + 0.389i)5-s + (0.996 − 0.0884i)7-s + (−0.874 + 0.505i)11-s − 1.32·13-s + (−1.13 + 0.655i)17-s + (−0.453 + 0.786i)19-s + (−0.986 − 0.569i)23-s + (−0.195 − 0.339i)25-s + 0.301i·29-s + (−0.710 − 1.23i)31-s + (0.707 + 0.328i)35-s + (−0.522 + 0.904i)37-s + 1.07i·41-s + 0.724·43-s + (−0.274 − 0.158i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.276i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.961 - 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5881002585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5881002585\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-6.97 + 0.619i)T \) |
| good | 5 | \( 1 + (-3.37 - 1.94i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (9.62 - 5.55i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 17.2T + 169T^{2} \) |
| 17 | \( 1 + (19.3 - 11.1i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (8.62 - 14.9i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (22.6 + 13.1i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 8.73iT - 841T^{2} \) |
| 31 | \( 1 + (22.0 + 38.1i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (19.3 - 33.4i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 44.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 31.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + (12.8 + 7.43i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-30.9 + 17.8i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (2.75 - 1.58i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-21.7 + 37.6i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (13.8 + 24.0i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 97.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (68.4 + 118. i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-27.9 + 48.4i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 137. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-44.6 - 25.7i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 16.6T + 9.40e3T^{2} \) |
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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
−10.24560607293157164657552241646, −9.491271194021208498584981205347, −8.293491962481217190306116076637, −7.77509625090970976201627640455, −6.76554971401165281837088824119, −5.85797275740573054410265464469, −4.90199566480296944143638243546, −4.13907325782251563742782785876, −2.41789613707191055112091512861, −1.94514443247068437421389005239,
0.16000386352794005135151068551, 1.86809501137886406685055096954, 2.62254009924682250867212801850, 4.25828246511529121438723005466, 5.16601471100497359801735337803, 5.63121972042069675798291310101, 7.02400440876745292391811822355, 7.66905372412188522576226457538, 8.728689810819364305826437633849, 9.239619243977567214031135157109
