L(s) = 1 | + (1 − 1.73i)5-s + (1.5 + 2.59i)7-s + (1 + 1.73i)11-s + (−0.5 + 0.866i)13-s + 8·17-s − 7·19-s + (−2 + 3.46i)23-s + (0.500 + 0.866i)25-s + (3 + 5.19i)29-s + (0.5 − 0.866i)31-s + 6·35-s + 3·37-s + (−3 + 5.19i)41-s + (−0.5 − 0.866i)43-s + (−3 − 5.19i)47-s + ⋯ |
L(s) = 1 | + (0.447 − 0.774i)5-s + (0.566 + 0.981i)7-s + (0.301 + 0.522i)11-s + (−0.138 + 0.240i)13-s + 1.94·17-s − 1.60·19-s + (−0.417 + 0.722i)23-s + (0.100 + 0.173i)25-s + (0.557 + 0.964i)29-s + (0.0898 − 0.155i)31-s + 1.01·35-s + 0.493·37-s + (−0.468 + 0.811i)41-s + (−0.0762 − 0.132i)43-s + (−0.437 − 0.757i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.033378323\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.033378323\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.5 - 2.59i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 8T + 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2 - 3.46i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 18T + 89T^{2} \) |
| 97 | \( 1 + (8.5 + 14.7i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.312523428308712104228176243031, −8.456683151332490839232268802458, −7.977511600850153739074533863088, −6.86670034555647860650359169089, −5.91683784638357110464508458588, −5.25521572721378184386647043572, −4.58867643915452877071653637900, −3.40311402988863608329388347933, −2.13026423261067566046519310695, −1.33571459212161838585594480491,
0.808038523639830061805570994212, 2.12723841924635981643273424738, 3.21488838268326878828764095273, 4.10008445752465729411140938516, 5.01566326428311205193651027398, 6.20742409324371474698932728431, 6.51261225722573665350274347850, 7.78289914537829508958637796739, 8.006387480769418463158438555176, 9.148691319394910927608324339266