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
−11.33285976945647931354172532444, −10.92291465349261202771275437907, −9.018059285395865892514036476536, −8.577607394702347207694896945275, −7.62147315879243082961269432454, −6.41670431197760104788318770904, −5.27360976301483682553482516462, −3.87663446263378525551488222362, −2.93450922018306239870251421392, −0.21583530554120625170383068118,
2.43505897350052590295783916355, 4.06259546390263778575907894733, 4.71955712558495904086712484447, 6.43763689212914654299157888023, 7.37462065629659951206515896210, 8.239115227098095093799876120609, 9.131960517314083419744867184222, 10.45102341689496523603628799846, 11.20994367507544354887047266744, 12.22473611831503623886072914171