| L(s) = 1 | − 3.27·5-s − 0.418i·7-s − 3·9-s + 6.50i·11-s − 7.27·17-s + 4.35i·19-s − 8.71i·23-s + 5.72·25-s + 1.37i·35-s − 5.67i·43-s + 9.82·45-s + 13.4i·47-s + 6.82·49-s − 21.3i·55-s − 11.2·61-s + ⋯ |
| L(s) = 1 | − 1.46·5-s − 0.158i·7-s − 9-s + 1.96i·11-s − 1.76·17-s + 0.999i·19-s − 1.81i·23-s + 1.14·25-s + 0.231i·35-s − 0.865i·43-s + 1.46·45-s + 1.96i·47-s + 0.974·49-s − 2.87i·55-s − 1.44·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0803858 + 0.300003i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0803858 + 0.300003i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 3.27T + 5T^{2} \) |
| 7 | \( 1 + 0.418iT - 7T^{2} \) |
| 11 | \( 1 - 6.50iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 7.27T + 17T^{2} \) |
| 23 | \( 1 + 8.71iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 5.67iT - 43T^{2} \) |
| 47 | \( 1 - 13.4iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 5.82T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 8.71iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−12.22473611831503623886072914171, −11.20994367507544354887047266744, −10.45102341689496523603628799846, −9.131960517314083419744867184222, −8.239115227098095093799876120609, −7.37462065629659951206515896210, −6.43763689212914654299157888023, −4.71955712558495904086712484447, −4.06259546390263778575907894733, −2.43505897350052590295783916355,
0.21583530554120625170383068118, 2.93450922018306239870251421392, 3.87663446263378525551488222362, 5.27360976301483682553482516462, 6.41670431197760104788318770904, 7.62147315879243082961269432454, 8.577607394702347207694896945275, 9.018059285395865892514036476536, 10.92291465349261202771275437907, 11.33285976945647931354172532444
