| L(s) = 1 | − 3-s + 4.33i·5-s + (−1.65 − 2.06i)7-s + 9-s + 3.79i·11-s − 2.82i·13-s − 4.33i·15-s + 4.33i·17-s − 2.54·19-s + (1.65 + 2.06i)21-s − 5.64i·23-s − 13.8·25-s − 27-s − 9.50·29-s − 1.84·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.93i·5-s + (−0.624 − 0.781i)7-s + 0.333·9-s + 1.14i·11-s − 0.784i·13-s − 1.11i·15-s + 1.05i·17-s − 0.582·19-s + (0.360 + 0.450i)21-s − 1.17i·23-s − 2.76·25-s − 0.192·27-s − 1.76·29-s − 0.331·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0274120 + 0.493712i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0274120 + 0.493712i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + (1.65 + 2.06i)T \) |
| good | 5 | \( 1 - 4.33iT - 5T^{2} \) |
| 11 | \( 1 - 3.79iT - 11T^{2} \) |
| 13 | \( 1 + 2.82iT - 13T^{2} \) |
| 17 | \( 1 - 4.33iT - 17T^{2} \) |
| 19 | \( 1 + 2.54T + 19T^{2} \) |
| 23 | \( 1 + 5.64iT - 23T^{2} \) |
| 29 | \( 1 + 9.50T + 29T^{2} \) |
| 31 | \( 1 + 1.84T + 31T^{2} \) |
| 37 | \( 1 - 5.11T + 37T^{2} \) |
| 41 | \( 1 + 1.32iT - 41T^{2} \) |
| 43 | \( 1 - 2.47iT - 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 - 1.23T + 53T^{2} \) |
| 59 | \( 1 - 1.65T + 59T^{2} \) |
| 61 | \( 1 + 3.50iT - 61T^{2} \) |
| 67 | \( 1 - 11.1iT - 67T^{2} \) |
| 71 | \( 1 - 3.03iT - 71T^{2} \) |
| 73 | \( 1 + 3.01iT - 73T^{2} \) |
| 79 | \( 1 - 10.1iT - 79T^{2} \) |
| 83 | \( 1 + 3.04T + 83T^{2} \) |
| 89 | \( 1 - 6.53iT - 89T^{2} \) |
| 97 | \( 1 + 6.33iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.78079677224267038533386491653, −10.23344521851819721421311704330, −9.692448403624868847978319543143, −7.975793626800910954297486606073, −7.17688319649107578008969602515, −6.59176483972666762884420776573, −5.84870191702487781398863531914, −4.27143451512024032757198290480, −3.40310535952134666379349972430, −2.16220883412885156634864926429,
0.27147903248093502454737934707, 1.76458186678987297445594492216, 3.58737312768157205997531490504, 4.76101203949050892821099632239, 5.53174976002617491887996552037, 6.16633784931076668344684475855, 7.55196685698261915806745258445, 8.619075731272395207198219031704, 9.216855221083478225832741828698, 9.733074169197089486985771087723
