L(s) = 1 | + (−1.28 + 0.595i)2-s + (−1.03 + 1.38i)3-s + (1.29 − 1.52i)4-s + 1.31·5-s + (0.501 − 2.39i)6-s − 1.36i·7-s + (−0.744 + 2.72i)8-s + (−0.855 − 2.87i)9-s + (−1.69 + 0.786i)10-s + 1.33i·11-s + (0.785 + 3.37i)12-s − 4.58i·13-s + (0.813 + 1.75i)14-s + (−1.36 + 1.83i)15-s + (−0.670 − 3.94i)16-s − 4.76i·17-s + ⋯ |
L(s) = 1 | + (−0.906 + 0.421i)2-s + (−0.597 + 0.801i)3-s + (0.645 − 0.764i)4-s + 0.590·5-s + (0.204 − 0.978i)6-s − 0.516i·7-s + (−0.263 + 0.964i)8-s + (−0.285 − 0.958i)9-s + (−0.535 + 0.248i)10-s + 0.403i·11-s + (0.226 + 0.973i)12-s − 1.27i·13-s + (0.217 + 0.468i)14-s + (−0.352 + 0.472i)15-s + (−0.167 − 0.985i)16-s − 1.15i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.615 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.520193 - 0.253591i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.520193 - 0.253591i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.28 - 0.595i)T \) |
| 3 | \( 1 + (1.03 - 1.38i)T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 1.31T + 5T^{2} \) |
| 7 | \( 1 + 1.36iT - 7T^{2} \) |
| 11 | \( 1 - 1.33iT - 11T^{2} \) |
| 13 | \( 1 + 4.58iT - 13T^{2} \) |
| 17 | \( 1 + 4.76iT - 17T^{2} \) |
| 19 | \( 1 + 6.45T + 19T^{2} \) |
| 29 | \( 1 + 4.15T + 29T^{2} \) |
| 31 | \( 1 + 8.95iT - 31T^{2} \) |
| 37 | \( 1 - 2.96iT - 37T^{2} \) |
| 41 | \( 1 + 6.40iT - 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 + 4.02T + 47T^{2} \) |
| 53 | \( 1 - 9.40T + 53T^{2} \) |
| 59 | \( 1 - 4.77iT - 59T^{2} \) |
| 61 | \( 1 + 13.3iT - 61T^{2} \) |
| 67 | \( 1 - 7.21T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 - 3.85T + 73T^{2} \) |
| 79 | \( 1 - 0.671iT - 79T^{2} \) |
| 83 | \( 1 - 6.46iT - 83T^{2} \) |
| 89 | \( 1 - 4.72iT - 89T^{2} \) |
| 97 | \( 1 + 5.63T + 97T^{2} \) |
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
−10.49427909965174145901619726268, −9.786426914606545147533918860376, −9.177866695537224402087238456074, −8.017283481290428942947976383586, −7.07054936302062539451397030384, −6.00445839341455205347345525919, −5.38396960338118411255537842731, −4.12588007143926384286666985789, −2.41403524465434971132900336808, −0.47061362666224218587507171097,
1.59106185910355983350217452823, 2.37475867885945440955963028746, 4.12546070250741615252840591371, 5.82080290173815757600298539179, 6.40377689567843571121661002308, 7.34340637478427095759196710334, 8.480710272972796114951493597573, 8.983310767058892970432322681958, 10.19132544621734080419302351782, 10.89947005829213208774653488565