| L(s) = 1 | + 2.23i·5-s − 0.246i·7-s − 3.93i·11-s − 16.2·13-s + 9.26i·17-s + 9.27i·19-s + (11.8 − 19.6i)23-s − 5.00·25-s + 12.7·29-s − 18.4·31-s + 0.551·35-s + 32.9i·37-s − 16.7·41-s − 71.8i·43-s − 72.2·47-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s − 0.0352i·7-s − 0.357i·11-s − 1.25·13-s + 0.545i·17-s + 0.488i·19-s + (0.516 − 0.856i)23-s − 0.200·25-s + 0.438·29-s − 0.595·31-s + 0.0157·35-s + 0.889i·37-s − 0.408·41-s − 1.67i·43-s − 1.53·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.520673148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.520673148\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (-11.8 + 19.6i)T \) |
| good | 7 | \( 1 + 0.246iT - 49T^{2} \) |
| 11 | \( 1 + 3.93iT - 121T^{2} \) |
| 13 | \( 1 + 16.2T + 169T^{2} \) |
| 17 | \( 1 - 9.26iT - 289T^{2} \) |
| 19 | \( 1 - 9.27iT - 361T^{2} \) |
| 29 | \( 1 - 12.7T + 841T^{2} \) |
| 31 | \( 1 + 18.4T + 961T^{2} \) |
| 37 | \( 1 - 32.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 16.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 71.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 72.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 8.57iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 68.8T + 3.48e3T^{2} \) |
| 61 | \( 1 - 5.62iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 72.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 100.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 115.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 0.109iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 27.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 59.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 160. iT - 9.40e3T^{2} \) |
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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
−8.285294362808778430872351480410, −7.29805878084730972696474546611, −6.86559950216750995982785067567, −5.98789084828732032386663710442, −5.22376171956649255305368475290, −4.41779522263137382432143061788, −3.49803132220147471901856866318, −2.68394668125718606638207516831, −1.79372159572215773272524881619, −0.43323573772500994230985414565,
0.70997209457837644970038174993, 1.89848520949261564449197696179, 2.76498133586818600968685012552, 3.72309010580621988663988507457, 4.87110167294480029194177417843, 5.02133945760840742864808131810, 6.09430399488236672647070273593, 7.02416902870483627387860914227, 7.49976670048757048015489706286, 8.268241172592359177424211933111
