| L(s) = 1 | + i·2-s − 4-s − 0.765i·5-s + (−2.29 − 1.32i)7-s − i·8-s + 0.765·10-s − 4.09·11-s + 1.23·13-s + (1.32 − 2.29i)14-s + 16-s + 3.96i·17-s + (3.29 − 2.84i)19-s + 0.765i·20-s − 4.09i·22-s − 2.23·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.342i·5-s + (−0.866 − 0.499i)7-s − 0.353i·8-s + 0.242·10-s − 1.23·11-s + 0.342·13-s + (0.353 − 0.612i)14-s + 0.250·16-s + 0.961i·17-s + (0.757 − 0.653i)19-s + 0.171i·20-s − 0.872i·22-s − 0.466·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.188 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.005505315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.005505315\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.29 + 1.32i)T \) |
| 19 | \( 1 + (-3.29 + 2.84i)T \) |
| good | 5 | \( 1 + 0.765iT - 5T^{2} \) |
| 11 | \( 1 + 4.09T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 3.96iT - 17T^{2} \) |
| 23 | \( 1 + 2.23T + 23T^{2} \) |
| 29 | \( 1 - 3.54iT - 29T^{2} \) |
| 31 | \( 1 + 8.42T + 31T^{2} \) |
| 37 | \( 1 + 5.83iT - 37T^{2} \) |
| 41 | \( 1 - 5.62T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 - 10.9iT - 47T^{2} \) |
| 53 | \( 1 + 8.28iT - 53T^{2} \) |
| 59 | \( 1 + 6.22T + 59T^{2} \) |
| 61 | \( 1 - 5.64iT - 61T^{2} \) |
| 67 | \( 1 - 13.7iT - 67T^{2} \) |
| 71 | \( 1 - 11.2iT - 71T^{2} \) |
| 73 | \( 1 - 9.01iT - 73T^{2} \) |
| 79 | \( 1 - 6.21iT - 79T^{2} \) |
| 83 | \( 1 - 2.03iT - 83T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 - 1.90T + 97T^{2} \) |
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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
−9.116458008718623363610115813781, −8.359848330974229330463892101668, −7.48098294587609812272780184928, −7.05387316721712871057825453015, −5.94341481270917855009744722869, −5.49370356159718271989253422256, −4.44746565945770843353023503620, −3.62842881085484322906023170866, −2.60167349508566344096925363618, −0.941412141436753737195520076390,
0.42586171595871940510973829737, 2.07253650997932498865400781480, 2.94040862873574511194046672964, 3.52415791421660547601241540660, 4.76003372951533509584666796029, 5.57028361966945700438624203504, 6.26859135519891705255851813544, 7.39993646296349372851416390284, 7.936444777077488009645501231537, 9.123801857850617080765035246913
