| L(s) = 1 | + (−3.41 − 5.91i)5-s + (14.9 − 10.9i)7-s + (−50.5 − 29.1i)11-s − 38.5i·13-s + (−16.1 + 27.9i)17-s + (−107. + 62.2i)19-s + (−174. + 100. i)23-s + (39.2 − 67.9i)25-s + 104. i·29-s + (240. + 138. i)31-s + (−115. − 50.9i)35-s + (23.8 + 41.2i)37-s + 387.·41-s − 272.·43-s + (−81.5 − 141. i)47-s + ⋯ |
| L(s) = 1 | + (−0.305 − 0.528i)5-s + (0.806 − 0.591i)7-s + (−1.38 − 0.799i)11-s − 0.822i·13-s + (−0.230 + 0.398i)17-s + (−1.30 + 0.751i)19-s + (−1.57 + 0.911i)23-s + (0.313 − 0.543i)25-s + 0.668i·29-s + (1.39 + 0.805i)31-s + (−0.558 − 0.245i)35-s + (0.105 + 0.183i)37-s + 1.47·41-s − 0.966·43-s + (−0.253 − 0.438i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.143 - 0.989i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.143 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5007801245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5007801245\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (-14.9 + 10.9i)T \) |
| good | 5 | \( 1 + (3.41 + 5.91i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (50.5 + 29.1i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 38.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (16.1 - 27.9i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (107. - 62.2i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (174. - 100. i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 104. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-240. - 138. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-23.8 - 41.2i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 387.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 272.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (81.5 + 141. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-313. - 181. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (105. - 183. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (202. - 117. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-262. + 454. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 348. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-465. - 268. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-362. - 628. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 392.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (430. + 744. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 978. iT - 9.12e5T^{2} \) |
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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
−10.17216319530775666070233419591, −8.569355427284482264331906458138, −8.202347443206060816712815578442, −7.64328014876872355629902701760, −6.29312533304544266501317353847, −5.42899221700183300403931007208, −4.57626150107904661089412484840, −3.65374291311348643061506242117, −2.36269841259893481644882709131, −1.03693477737668297453739621535,
0.13454894418261246178983100699, 2.11883087752529687761457754884, 2.57429718181268971484786199900, 4.27742722578591154458186557471, 4.78212433412927965083609692749, 5.98550308203713896668224850652, 6.84723974220290179644924209093, 7.82313484454120123615336461561, 8.334000939950050925740056446347, 9.382650576425237948786416864104
