L(s) = 1 | + (3.23 − 3.81i)5-s − 12.6i·7-s − 18.7i·11-s − 6.79i·13-s + 11.5·17-s + 24.2·19-s + 29.2·23-s + (−4.04 − 24.6i)25-s + 14.9i·29-s + 13.8·31-s + (−48.2 − 40.9i)35-s + 38.2i·37-s − 30.9i·41-s − 13.0i·43-s + 5.68·47-s + ⋯ |
L(s) = 1 | + (0.647 − 0.762i)5-s − 1.80i·7-s − 1.70i·11-s − 0.522i·13-s + 0.679·17-s + 1.27·19-s + 1.27·23-s + (−0.161 − 0.986i)25-s + 0.515i·29-s + 0.446·31-s + (−1.37 − 1.17i)35-s + 1.03i·37-s − 0.754i·41-s − 0.304i·43-s + 0.121·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.647 + 0.762i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.647 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.494607721\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.494607721\) |
\(L(2)\) |
|
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 | \( 1 + (-3.23 + 3.81i)T \) |
good | 7 | \( 1 + 12.6iT - 49T^{2} \) |
| 11 | \( 1 + 18.7iT - 121T^{2} \) |
| 13 | \( 1 + 6.79iT - 169T^{2} \) |
| 17 | \( 1 - 11.5T + 289T^{2} \) |
| 19 | \( 1 - 24.2T + 361T^{2} \) |
| 23 | \( 1 - 29.2T + 529T^{2} \) |
| 29 | \( 1 - 14.9iT - 841T^{2} \) |
| 31 | \( 1 - 13.8T + 961T^{2} \) |
| 37 | \( 1 - 38.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 30.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 13.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 5.68T + 2.20e3T^{2} \) |
| 53 | \( 1 + 57.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 30.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 77.0T + 3.72e3T^{2} \) |
| 67 | \( 1 - 128. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 35.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 40.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 140.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 118.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 75.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 84.5iT - 9.40e3T^{2} \) |
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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
−8.893261592915050209438781141546, −8.135072985243331224897115636634, −7.39169647192220458766336485655, −6.49732144886405554578834831837, −5.52691459781763616551726847857, −4.90265911913298656370893161649, −3.70881710341506373997830833685, −3.03898166280921894964771625269, −1.08900926240027644592558779932, −0.822747751648053040773236274998,
1.66169519137407112917641278686, 2.44720253004712104113722274821, 3.24457486821618830120439559023, 4.77583293951761942263681783918, 5.41550581970998111036330156628, 6.24579133860059724115955383621, 7.04698493874900065805496194344, 7.80546808998222299488223449520, 8.948136991385211273413429402823, 9.632338544151302499091675187066