| L(s) = 1 | − 1.41i·2-s − 3-s − 2.00·4-s + 1.41i·6-s + 8.48i·7-s + 2.82i·8-s − 8·9-s + (7 + 8.48i)11-s + 2.00·12-s − 8.48i·13-s + 12·14-s + 4.00·16-s − 25.4i·17-s + 11.3i·18-s − 25.4i·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.333·3-s − 0.500·4-s + 0.235i·6-s + 1.21i·7-s + 0.353i·8-s − 0.888·9-s + (0.636 + 0.771i)11-s + 0.166·12-s − 0.652i·13-s + 0.857·14-s + 0.250·16-s − 1.49i·17-s + 0.628i·18-s − 1.33i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7992134294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7992134294\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-7 - 8.48i)T \) |
| good | 3 | \( 1 + T + 9T^{2} \) |
| 7 | \( 1 - 8.48iT - 49T^{2} \) |
| 13 | \( 1 + 8.48iT - 169T^{2} \) |
| 17 | \( 1 + 25.4iT - 289T^{2} \) |
| 19 | \( 1 + 25.4iT - 361T^{2} \) |
| 23 | \( 1 + 17T + 529T^{2} \) |
| 29 | \( 1 + 33.9iT - 841T^{2} \) |
| 31 | \( 1 - 17T + 961T^{2} \) |
| 37 | \( 1 + 47T + 1.36e3T^{2} \) |
| 41 | \( 1 - 8.48iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 16.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 58T + 2.20e3T^{2} \) |
| 53 | \( 1 + 2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 55T + 3.48e3T^{2} \) |
| 61 | \( 1 + 84.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 89T + 4.48e3T^{2} \) |
| 71 | \( 1 + 7T + 5.04e3T^{2} \) |
| 73 | \( 1 + 127. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 33.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 33.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 97T + 7.92e3T^{2} \) |
| 97 | \( 1 - 121T + 9.40e3T^{2} \) |
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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
−10.29844631800142809338170271274, −9.279868907307245706834166775879, −8.853403449351063063507526591558, −7.66416152340962625399428021614, −6.39668900968460893681824980723, −5.42034299918719778459836369559, −4.63585754970875912102820291767, −3.05009044123995312551541275156, −2.23851789080031646694036441241, −0.34380741430907695462233130714,
1.32797959855306721159989798168, 3.53436877559430937119423676310, 4.25619744154938485020322118544, 5.69318296671146651417641921757, 6.28112589983982775409990639499, 7.24154147383847849549567517273, 8.285844285297308456968477553250, 8.871855166252757322571568332013, 10.21147180372204068866510122431, 10.73406216707013856330998945468
