L(s) = 1 | − 4.33i·5-s + (−1.65 + 2.06i)7-s − 3.79i·11-s − 2.82i·13-s + 4.33i·17-s + 2.54·19-s − 5.64i·23-s − 13.8·25-s − 9.50·29-s − 1.84·31-s + (8.96 + 7.16i)35-s − 5.11·37-s − 1.32i·41-s + 2.47i·43-s + 12.2·47-s + ⋯ |
L(s) = 1 | − 1.93i·5-s + (−0.624 + 0.781i)7-s − 1.14i·11-s − 0.784i·13-s + 1.05i·17-s + 0.582·19-s − 1.17i·23-s − 2.76·25-s − 1.76·29-s − 0.331·31-s + (1.51 + 1.21i)35-s − 0.841·37-s − 0.206i·41-s + 0.377i·43-s + 1.78·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5059034072\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5059034072\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 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 | \( 1 - 6.33iT - 97T^{2} \) |
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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
−8.304367182641148060026103475680, −7.47524971638027354730027304933, −6.19665661593876212351827157758, −5.60012696881273525385902017987, −5.28738038811970931650190798718, −4.13477826915223024012536616778, −3.46398947515967046722207606005, −2.26893126124055109152378469379, −1.13513606457167225525062691265, −0.15072791298593350011684646453,
1.77546544740967559892489179263, 2.64903575101873700121209939988, 3.54610929969177486430406956496, 4.01529952921820906815227438375, 5.24753746985872893560790853162, 6.14059785069707345232889788254, 6.99974694259246704372667321769, 7.26763545015750851556729090326, 7.61867196741052517589330127985, 9.314777967670679266089649895431