| 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
−9.239619243977567214031135157109, −8.728689810819364305826437633849, −7.66905372412188522576226457538, −7.02400440876745292391811822355, −5.63121972042069675798291310101, −5.16601471100497359801735337803, −4.25828246511529121438723005466, −2.62254009924682250867212801850, −1.86809501137886406685055096954, −0.16000386352794005135151068551,
1.94514443247068437421389005239, 2.41789613707191055112091512861, 4.13907325782251563742782785876, 4.90199566480296944143638243546, 5.85797275740573054410265464469, 6.76554971401165281837088824119, 7.77509625090970976201627640455, 8.293491962481217190306116076637, 9.491271194021208498584981205347, 10.24560607293157164657552241646
