| L(s) = 1 | + (−2.74 − 1.58i)5-s + (2.24 + 6.63i)7-s + (6.62 + 11.4i)11-s − 5.49i·13-s + (11.7 − 6.77i)17-s + (0.621 + 0.358i)19-s + (1.13 − 1.96i)23-s + (−7.48 − 12.9i)25-s − 20.4·29-s + (−21.3 + 12.3i)31-s + (4.34 − 21.7i)35-s + (−32.4 + 56.2i)37-s + 21.0i·41-s − 6.48·43-s + (41.3 + 23.8i)47-s + ⋯ |
| L(s) = 1 | + (−0.548 − 0.316i)5-s + (0.320 + 0.947i)7-s + (0.601 + 1.04i)11-s − 0.422i·13-s + (0.690 − 0.398i)17-s + (0.0327 + 0.0188i)19-s + (0.0493 − 0.0855i)23-s + (−0.299 − 0.518i)25-s − 0.706·29-s + (−0.687 + 0.397i)31-s + (0.124 − 0.621i)35-s + (−0.877 + 1.52i)37-s + 0.512i·41-s − 0.150·43-s + (0.880 + 0.508i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.321071665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.321071665\) |
| \(L(2)\) |
|
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 | \( 1 + (-2.24 - 6.63i)T \) |
| good | 5 | \( 1 + (2.74 + 1.58i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (-6.62 - 11.4i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 5.49iT - 169T^{2} \) |
| 17 | \( 1 + (-11.7 + 6.77i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-0.621 - 0.358i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-1.13 + 1.96i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 20.4T + 841T^{2} \) |
| 31 | \( 1 + (21.3 - 12.3i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (32.4 - 56.2i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 21.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 6.48T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-41.3 - 23.8i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-11.0 - 19.0i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (72.5 - 41.8i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-57.3 - 33.1i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-46.3 - 80.2i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 48.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-113. + 65.4i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (38.1 - 66.0i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 107. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-145. - 83.9i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 25.5iT - 9.40e3T^{2} \) |
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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
−9.858839754287081367049786808345, −9.177373821702248553792883029806, −8.322451810417608101138399532148, −7.61388889304240139590498802844, −6.67570384973563809576387158447, −5.56984056293077494805925301080, −4.81987667928115259172383779581, −3.81455847275747412887885291155, −2.59305724571531226616796798388, −1.34612373525956521621504541389,
0.43932548755545809137213541239, 1.78975212288291092939756399313, 3.59992664535128905924009532144, 3.79668318118551580178035468472, 5.19734247241271817370178921918, 6.16529502360746476967197292153, 7.19301362958897825068699677004, 7.71128833517156111161653371084, 8.691925220655708864995217761252, 9.494921814911949876961509980078
