L(s) = 1 | + (−0.0745 − 1.73i)3-s + (−2.11 + 3.66i)5-s + (3.25 − 1.87i)7-s + (−2.98 + 0.258i)9-s + (0.424 − 0.245i)11-s + (1.78 + 1.03i)13-s + (6.49 + 3.38i)15-s + 4.97i·17-s − 6.32·19-s + (−3.49 − 5.49i)21-s + (−0.0297 + 0.0514i)23-s + (−6.44 − 11.1i)25-s + (0.669 + 5.15i)27-s + (2.80 + 4.86i)29-s + (0.546 + 0.315i)31-s + ⋯ |
L(s) = 1 | + (−0.0430 − 0.999i)3-s + (−0.946 + 1.63i)5-s + (1.23 − 0.710i)7-s + (−0.996 + 0.0860i)9-s + (0.128 − 0.0739i)11-s + (0.496 + 0.286i)13-s + (1.67 + 0.874i)15-s + 1.20i·17-s − 1.45·19-s + (−0.762 − 1.19i)21-s + (−0.00619 + 0.0107i)23-s + (−1.28 − 2.23i)25-s + (0.128 + 0.991i)27-s + (0.521 + 0.903i)29-s + (0.0981 + 0.0566i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.202878907\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.202878907\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.0745 + 1.73i)T \) |
good | 5 | \( 1 + (2.11 - 3.66i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-3.25 + 1.87i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.424 + 0.245i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.78 - 1.03i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 4.97iT - 17T^{2} \) |
| 19 | \( 1 + 6.32T + 19T^{2} \) |
| 23 | \( 1 + (0.0297 - 0.0514i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.80 - 4.86i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.546 - 0.315i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 6.74iT - 37T^{2} \) |
| 41 | \( 1 + (-4.44 - 2.56i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.35 - 7.53i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.58 - 7.93i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 1.32T + 53T^{2} \) |
| 59 | \( 1 + (6.67 + 3.85i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.56 + 1.48i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.07 - 3.59i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4.06T + 71T^{2} \) |
| 73 | \( 1 - 8.03T + 73T^{2} \) |
| 79 | \( 1 + (0.601 - 0.347i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.46 + 4.88i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 6.19iT - 89T^{2} \) |
| 97 | \( 1 + (0.864 + 1.49i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.40673075549759167160492074614, −8.720591585270550551187581844027, −7.948795526516592345766996811764, −7.62239771080371433967660715511, −6.53077734087791239222168966051, −6.27810279456471564512012427762, −4.56333149863109324799158563193, −3.71175970266137912601159098889, −2.59345608010481736902527152218, −1.39720409829348193729663215506,
0.56372902624290302794932239373, 2.24009310005877568690144477921, 3.87372150809158241943468472652, 4.49605787464431542175602945451, 5.14290763741408268970746808376, 5.85897311439689520285665721641, 7.55193173281981447881054613942, 8.350375477216174765998497373629, 8.794832039629473811594663698951, 9.330843848213808092835326327777