| L(s) = 1 | − 1.90i·2-s − 1.62·4-s + (−0.311 − 2.21i)5-s + i·7-s − 0.719i·8-s + (−4.21 + 0.592i)10-s − 2·11-s − 6.42i·13-s + 1.90·14-s − 4.61·16-s + 4.42i·17-s + 2.42·19-s + (0.504 + 3.59i)20-s + 3.80i·22-s − 1.37i·23-s + ⋯ |
| L(s) = 1 | − 1.34i·2-s − 0.811·4-s + (−0.139 − 0.990i)5-s + 0.377i·7-s − 0.254i·8-s + (−1.33 + 0.187i)10-s − 0.603·11-s − 1.78i·13-s + 0.508·14-s − 1.15·16-s + 1.07i·17-s + 0.557·19-s + (0.112 + 0.803i)20-s + 0.811i·22-s − 0.287i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0807292 - 1.15482i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0807292 - 1.15482i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.311 + 2.21i)T \) |
| 7 | \( 1 - iT \) |
| good | 2 | \( 1 + 1.90iT - 2T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 6.42iT - 13T^{2} \) |
| 17 | \( 1 - 4.42iT - 17T^{2} \) |
| 19 | \( 1 - 2.42T + 19T^{2} \) |
| 23 | \( 1 + 1.37iT - 23T^{2} \) |
| 29 | \( 1 - 0.755T + 29T^{2} \) |
| 31 | \( 1 - 5.18T + 31T^{2} \) |
| 37 | \( 1 + 7.61iT - 37T^{2} \) |
| 41 | \( 1 - 8.23T + 41T^{2} \) |
| 43 | \( 1 - 10.1iT - 43T^{2} \) |
| 47 | \( 1 - 2.75iT - 47T^{2} \) |
| 53 | \( 1 + 9.18iT - 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 - 6.85T + 61T^{2} \) |
| 67 | \( 1 + 2.75iT - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 1.57iT - 73T^{2} \) |
| 79 | \( 1 - 4.85T + 79T^{2} \) |
| 83 | \( 1 + 11.6iT - 83T^{2} \) |
| 89 | \( 1 - 4.62T + 89T^{2} \) |
| 97 | \( 1 - 11.9iT - 97T^{2} \) |
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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
−11.23634399219264973760496959229, −10.37724746837509704687809318791, −9.650483375477971095456722590059, −8.530731779781006143321486675667, −7.75833569721848916016236654672, −5.96417519311501408671529703200, −4.92992100142721254911065845139, −3.65950212843545944732624227888, −2.46289224517229816143212144231, −0.860010997950586480453610133078,
2.54434295755822099227426245685, 4.20842191549375276953173246011, 5.39251499186880952926456768582, 6.63602379162579032305666917481, 7.07937436011792164460390027537, 7.938948381262878704563636426384, 9.107043092047783664133749077022, 10.10655980499516049302005459486, 11.29576669918162381250178224716, 11.84242126176842143682160253316
