L(s) = 1 | + (−0.990 + 1.00i)2-s + (−0.800 + 1.53i)3-s + (−0.0363 − 1.99i)4-s + (3.73 + 1.00i)5-s + (−0.757 − 2.32i)6-s + (−1.68 + 2.91i)7-s + (2.05 + 1.94i)8-s + (−1.71 − 2.45i)9-s + (−4.70 + 2.77i)10-s + (0.211 − 0.0566i)11-s + (3.10 + 1.54i)12-s + (−2.71 − 0.727i)13-s + (−1.27 − 4.58i)14-s + (−4.52 + 4.93i)15-s + (−3.99 + 0.145i)16-s + 4.23i·17-s + ⋯ |
L(s) = 1 | + (−0.700 + 0.713i)2-s + (−0.461 + 0.886i)3-s + (−0.0181 − 0.999i)4-s + (1.67 + 0.447i)5-s + (−0.309 − 0.951i)6-s + (−0.635 + 1.10i)7-s + (0.726 + 0.687i)8-s + (−0.573 − 0.819i)9-s + (−1.48 + 0.878i)10-s + (0.0637 − 0.0170i)11-s + (0.895 + 0.445i)12-s + (−0.753 − 0.201i)13-s + (−0.340 − 1.22i)14-s + (−1.16 + 1.27i)15-s + (−0.999 + 0.0363i)16-s + 1.02i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.367156 + 0.698721i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.367156 + 0.698721i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.990 - 1.00i)T \) |
| 3 | \( 1 + (0.800 - 1.53i)T \) |
good | 5 | \( 1 + (-3.73 - 1.00i)T + (4.33 + 2.5i)T^{2} \) |
| 7 | \( 1 + (1.68 - 2.91i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.211 + 0.0566i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (2.71 + 0.727i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 - 4.23iT - 17T^{2} \) |
| 19 | \( 1 + (1.12 + 1.12i)T + 19iT^{2} \) |
| 23 | \( 1 + (-3.33 + 1.92i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.03 + 0.545i)T + (25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (-7.21 + 4.16i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.66 + 2.66i)T + 37iT^{2} \) |
| 41 | \( 1 + (-1.70 - 2.95i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.25 + 4.68i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (2.34 - 4.07i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.58 + 7.58i)T - 53iT^{2} \) |
| 59 | \( 1 + (1.43 - 5.34i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (2.33 + 8.69i)T + (-52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (-1.38 + 5.17i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 7.53iT - 71T^{2} \) |
| 73 | \( 1 - 3.22iT - 73T^{2} \) |
| 79 | \( 1 + (-4.98 - 2.87i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.20 - 4.50i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 - 2.96T + 89T^{2} \) |
| 97 | \( 1 + (7.63 - 13.2i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−13.71721557174587199436119827128, −12.44191311139510994085624680684, −10.91847226723341408283714930385, −10.04282481187348562580982141647, −9.506932351722035447443499216086, −8.620814335613272308249785934638, −6.56767973727411199032656195356, −5.99046787237029157077738666803, −5.06579228382206163992886604698, −2.50278069165753680827078407150,
1.14321988557430552495786442776, 2.64577141399610779011878069013, 4.93776024989797545295092453686, 6.51543572122966010298790242501, 7.31664731849060580306718477400, 8.824479635884355203399544082721, 9.883978839606915034771189367312, 10.46606266457231588115009601307, 11.81352122578504721520084444570, 12.76639007245847927115773692263