| L(s) = 1 | + (2.81 + 2.04i)2-s + (1.27 + 3.91i)4-s + (−9.28 − 6.22i)5-s + 12.5·7-s + (4.17 − 12.8i)8-s + (−13.4 − 36.5i)10-s + (30.5 + 22.2i)11-s + (25.8 − 18.7i)13-s + (35.2 + 25.5i)14-s + (64.7 − 47.0i)16-s + (5.81 − 17.8i)17-s + (49.5 − 152. i)19-s + (12.5 − 44.3i)20-s + (40.6 + 125. i)22-s + (86.4 + 62.7i)23-s + ⋯ |
| L(s) = 1 | + (0.995 + 0.723i)2-s + (0.159 + 0.489i)4-s + (−0.830 − 0.556i)5-s + 0.675·7-s + (0.184 − 0.567i)8-s + (−0.424 − 1.15i)10-s + (0.837 + 0.608i)11-s + (0.551 − 0.400i)13-s + (0.672 + 0.488i)14-s + (1.01 − 0.734i)16-s + (0.0829 − 0.255i)17-s + (0.598 − 1.84i)19-s + (0.140 − 0.495i)20-s + (0.393 + 1.21i)22-s + (0.783 + 0.569i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0495i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.91826 - 0.0723198i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.91826 - 0.0723198i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (9.28 + 6.22i)T \) |
| good | 2 | \( 1 + (-2.81 - 2.04i)T + (2.47 + 7.60i)T^{2} \) |
| 7 | \( 1 - 12.5T + 343T^{2} \) |
| 11 | \( 1 + (-30.5 - 22.2i)T + (411. + 1.26e3i)T^{2} \) |
| 13 | \( 1 + (-25.8 + 18.7i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-5.81 + 17.8i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-49.5 + 152. i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-86.4 - 62.7i)T + (3.75e3 + 1.15e4i)T^{2} \) |
| 29 | \( 1 + (33.1 + 102. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (2.18 - 6.73i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (52.3 - 38.0i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (305. - 222. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 + 234.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (32.9 + 101. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-64.0 - 196. i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-261. + 190. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-221. - 161. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + (43.7 - 134. i)T + (-2.43e5 - 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-185. - 570. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-116. - 84.9i)T + (1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-98.8 - 304. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-344. + 1.05e3i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + (43.7 + 31.7i)T + (2.17e5 + 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-406. - 1.25e3i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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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
−11.85300831710331903211139024594, −11.26412610865721731938599970091, −9.707545914800301182051806495932, −8.638364824708497087769588122426, −7.48216273083261478781805740438, −6.71047002868862749085678667072, −5.21558986867795509471242686295, −4.62359966706116419622056663647, −3.46464408229364005105634061719, −1.04654557290215355018471111415,
1.59833677864551539951911296598, 3.34617403876803135298316942948, 3.94732983259767392701447825425, 5.22972434081489217093171417402, 6.53554178519210613046272184593, 7.902622116049468442232922817106, 8.676599898977218157953504196872, 10.38210256073664571924098788472, 11.20621875002671811979304546842, 11.79829614935202198567774836313
