L(s) = 1 | − 4.89i·3-s + 4.89i·5-s + (5 + 4.89i)7-s − 14.9·9-s − 6·11-s − 4.89i·13-s + 23.9·15-s + 19.5i·17-s − 24.4i·19-s + (23.9 − 24.4i)21-s − 30·23-s + 1.00·25-s + 29.3i·27-s − 6·29-s + 29.3i·33-s + ⋯ |
L(s) = 1 | − 1.63i·3-s + 0.979i·5-s + (0.714 + 0.699i)7-s − 1.66·9-s − 0.545·11-s − 0.376i·13-s + 1.59·15-s + 1.15i·17-s − 1.28i·19-s + (1.14 − 1.16i)21-s − 1.30·23-s + 0.0400·25-s + 1.08i·27-s − 0.206·29-s + 0.890i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.896934 - 0.366172i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.896934 - 0.366172i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-5 - 4.89i)T \) |
good | 3 | \( 1 + 4.89iT - 9T^{2} \) |
| 5 | \( 1 - 4.89iT - 25T^{2} \) |
| 11 | \( 1 + 6T + 121T^{2} \) |
| 13 | \( 1 + 4.89iT - 169T^{2} \) |
| 17 | \( 1 - 19.5iT - 289T^{2} \) |
| 19 | \( 1 + 24.4iT - 361T^{2} \) |
| 23 | \( 1 + 30T + 529T^{2} \) |
| 29 | \( 1 + 6T + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 10T + 1.36e3T^{2} \) |
| 41 | \( 1 + 48.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 10T + 1.84e3T^{2} \) |
| 47 | \( 1 - 19.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 90T + 2.80e3T^{2} \) |
| 59 | \( 1 - 24.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 24.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 70T + 4.48e3T^{2} \) |
| 71 | \( 1 - 42T + 5.04e3T^{2} \) |
| 73 | \( 1 - 107. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 74T + 6.24e3T^{2} \) |
| 83 | \( 1 + 63.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 146. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 78.3iT - 9.40e3T^{2} \) |
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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
−17.49546827430041657887027019916, −15.36279602132048395755420746153, −14.28217893432124476374902789828, −13.10212930056448555387090013297, −11.97016493292561868322426830582, −10.76355883241477193299422174829, −8.414074673860621653697820224068, −7.28194112582239693475608969263, −5.92187780477507538380273482265, −2.35788931264915256215796178855,
4.14483405066882728762060739027, 5.23974565656898765763989074898, 8.102851334220104601956306921892, 9.482227615847877260948823674436, 10.54152099235473083826173450415, 11.88210003694274100340271822057, 13.74088378107400828707586943021, 14.88408523470219201138873081677, 16.32510798008001579172429449035, 16.55687662739985045994637523519