| L(s) = 1 | + (0.804 + 1.16i)2-s + (−1.58 − 0.707i)3-s + (−0.705 + 1.87i)4-s + (0.422 − 0.0556i)5-s + (−0.449 − 2.40i)6-s + (−3.08 + 0.825i)7-s + (−2.74 + 0.685i)8-s + (1.99 + 2.23i)9-s + (0.404 + 0.446i)10-s + (−3.63 + 2.78i)11-s + (2.43 − 2.46i)12-s + (−0.682 + 0.889i)13-s + (−3.44 − 2.91i)14-s + (−0.707 − 0.211i)15-s + (−3.00 − 2.63i)16-s + 1.63i·17-s + ⋯ |
| L(s) = 1 | + (0.568 + 0.822i)2-s + (−0.912 − 0.408i)3-s + (−0.352 + 0.935i)4-s + (0.189 − 0.0248i)5-s + (−0.183 − 0.983i)6-s + (−1.16 + 0.312i)7-s + (−0.970 + 0.242i)8-s + (0.666 + 0.745i)9-s + (0.128 + 0.141i)10-s + (−1.09 + 0.840i)11-s + (0.703 − 0.710i)12-s + (−0.189 + 0.246i)13-s + (−0.919 − 0.780i)14-s + (−0.182 − 0.0544i)15-s + (−0.751 − 0.659i)16-s + 0.396i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0495091 + 0.660780i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0495091 + 0.660780i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.804 - 1.16i)T \) |
| 3 | \( 1 + (1.58 + 0.707i)T \) |
| good | 5 | \( 1 + (-0.422 + 0.0556i)T + (4.82 - 1.29i)T^{2} \) |
| 7 | \( 1 + (3.08 - 0.825i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (3.63 - 2.78i)T + (2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + (0.682 - 0.889i)T + (-3.36 - 12.5i)T^{2} \) |
| 17 | \( 1 - 1.63iT - 17T^{2} \) |
| 19 | \( 1 + (0.155 + 0.374i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-3.65 - 0.980i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-1.19 + 9.07i)T + (-28.0 - 7.50i)T^{2} \) |
| 31 | \( 1 + (-3.54 - 6.14i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.11 - 7.51i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-8.46 - 2.26i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (1.41 - 1.08i)T + (11.1 - 41.5i)T^{2} \) |
| 47 | \( 1 + (-3.74 - 2.16i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.78 - 2.80i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-7.91 + 1.04i)T + (56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (0.512 - 3.89i)T + (-58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (10.7 + 8.22i)T + (17.3 + 64.7i)T^{2} \) |
| 71 | \( 1 + (0.796 + 0.796i)T + 71iT^{2} \) |
| 73 | \( 1 + (2.82 - 2.82i)T - 73iT^{2} \) |
| 79 | \( 1 + (-8.74 - 5.05i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (14.3 + 1.88i)T + (80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (0.770 + 0.770i)T + 89iT^{2} \) |
| 97 | \( 1 + (9.05 - 15.6i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.45382348769342332952135045478, −11.71883211449785464181565662226, −10.32056659790465786429552028615, −9.480305418069965211818852495881, −8.045928984229526967939399809181, −7.11139988174386806229987480759, −6.28553392820645348357270850514, −5.45084615755479357616062677451, −4.38916080247231862392073383897, −2.66411679403254364436459817185,
0.44022550472303421336109842438, 2.84047071678189320111279768512, 3.95693241186632843280244347915, 5.29826804249441860461758212685, 5.95486504913958837462121152302, 7.10140984452864279645463200178, 8.952291753677550702967324955933, 9.937259366105038181079205373867, 10.51639497971029812955856679441, 11.26734785419452095591013226388
