L(s) = 1 | + (−1.93 − 1.11i)5-s + (−2.74 − 4.75i)7-s + (7.94 − 4.58i)11-s + (−5.74 + 9.94i)13-s + 16.9i·17-s + 26.9·19-s + (−4.27 − 2.46i)23-s + (2.5 + 4.33i)25-s + (−17.7 + 10.2i)29-s + (−10.4 + 18.1i)31-s + 12.2i·35-s − 62.4·37-s + (35.4 + 20.4i)41-s + (−0.513 − 0.888i)43-s + (74.6 − 43.1i)47-s + ⋯ |
L(s) = 1 | + (−0.387 − 0.223i)5-s + (−0.391 − 0.678i)7-s + (0.722 − 0.416i)11-s + (−0.441 + 0.765i)13-s + 0.998i·17-s + 1.41·19-s + (−0.185 − 0.107i)23-s + (0.100 + 0.173i)25-s + (−0.612 + 0.353i)29-s + (−0.338 + 0.585i)31-s + 0.350i·35-s − 1.68·37-s + (0.864 + 0.499i)41-s + (−0.0119 − 0.0206i)43-s + (1.58 − 0.917i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.693308552\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.693308552\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.93 + 1.11i)T \) |
good | 7 | \( 1 + (2.74 + 4.75i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-7.94 + 4.58i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (5.74 - 9.94i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 16.9iT - 289T^{2} \) |
| 19 | \( 1 - 26.9T + 361T^{2} \) |
| 23 | \( 1 + (4.27 + 2.46i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (17.7 - 10.2i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (10.4 - 18.1i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 62.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-35.4 - 20.4i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (0.513 + 0.888i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-74.6 + 43.1i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 96.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-97.3 - 56.1i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-33.4 - 57.9i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-38 + 65.8i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 24.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 18.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + (53.4 + 92.5i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-39.1 + 22.5i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 115. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-43.5 - 75.3i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−9.061687355235710128290788017261, −8.521657553733082458683283891330, −7.34152607601417114835362662411, −6.98776403530723190119035144392, −5.92711043987710945768349682364, −5.00627562773510862287258703660, −3.88693714812184564419958992234, −3.46820454802401170536619750071, −1.87890952806688220399252081192, −0.70381405719877988101267905707,
0.75902945607078682776009434895, 2.31633340594718057239299056996, 3.18987509467134113428894112702, 4.13362174230296697219546635515, 5.29446796313020055723145864157, 5.85785185436169778172587519251, 7.13442269269704812843328624978, 7.41262830534563124742075689488, 8.512544240766664351700999176486, 9.441895452467662710616756045927