L(s) = 1 | + (−2.44 − 2.44i)5-s + 1.41i·7-s + (−3.46 − 3.46i)11-s + (1 − i)13-s + 4.89·17-s + (−4.24 + 4.24i)19-s − 6.92i·23-s + 6.99i·25-s + (−2.44 + 2.44i)29-s − 1.41·31-s + (3.46 − 3.46i)35-s + (−7 − 7i)37-s + 4.89i·41-s + (4.24 + 4.24i)43-s − 6.92·47-s + ⋯ |
L(s) = 1 | + (−1.09 − 1.09i)5-s + 0.534i·7-s + (−1.04 − 1.04i)11-s + (0.277 − 0.277i)13-s + 1.18·17-s + (−0.973 + 0.973i)19-s − 1.44i·23-s + 1.39i·25-s + (−0.454 + 0.454i)29-s − 0.254·31-s + (0.585 − 0.585i)35-s + (−1.15 − 1.15i)37-s + 0.765i·41-s + (0.646 + 0.646i)43-s − 1.01·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2238698913\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2238698913\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.44 + 2.44i)T + 5iT^{2} \) |
| 7 | \( 1 - 1.41iT - 7T^{2} \) |
| 11 | \( 1 + (3.46 + 3.46i)T + 11iT^{2} \) |
| 13 | \( 1 + (-1 + i)T - 13iT^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 + (4.24 - 4.24i)T - 19iT^{2} \) |
| 23 | \( 1 + 6.92iT - 23T^{2} \) |
| 29 | \( 1 + (2.44 - 2.44i)T - 29iT^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 + (7 + 7i)T + 37iT^{2} \) |
| 41 | \( 1 - 4.89iT - 41T^{2} \) |
| 43 | \( 1 + (-4.24 - 4.24i)T + 43iT^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + (-2.44 - 2.44i)T + 53iT^{2} \) |
| 59 | \( 1 + (-6.92 - 6.92i)T + 59iT^{2} \) |
| 61 | \( 1 + (-5 + 5i)T - 61iT^{2} \) |
| 67 | \( 1 + (5.65 - 5.65i)T - 67iT^{2} \) |
| 71 | \( 1 - 13.8iT - 71T^{2} \) |
| 73 | \( 1 - 12iT - 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 + (10.3 - 10.3i)T - 83iT^{2} \) |
| 89 | \( 1 + 9.79iT - 89T^{2} \) |
| 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
−8.875712198241705735921228395675, −8.362964278224228443361657150255, −8.079679284826961922877572512046, −7.09687838144538795921121242077, −5.74792969321768094984243345268, −5.48500341183134388321516695976, −4.37312652307390390767855089320, −3.62848788395770994969054255579, −2.61591871942637526078275868087, −1.09142847567484563055834876916,
0.086505944238678698773967720629, 1.92378824447978559382523312153, 3.08332338823339124255941269776, 3.75694209941787233045555702805, 4.63621718623154535635362524624, 5.58224966586224867425127436789, 6.75526180773341904462654418215, 7.33135686111507890104789530828, 7.70230474492170418615931859333, 8.607824775731571836544678384435