L(s) = 1 | + (1.93 − 1.11i)5-s + (2 − 3.46i)7-s + (5.80 + 3.35i)11-s + (3.5 + 6.06i)13-s − 20.1i·17-s + 8·19-s + (−5.80 + 3.35i)23-s + (2.5 − 4.33i)25-s + (40.6 + 23.4i)29-s + (−14.5 − 25.1i)31-s − 8.94i·35-s + 2·37-s + (−11.6 + 6.70i)41-s + (3.5 − 6.06i)43-s + (29.0 + 16.7i)47-s + ⋯ |
L(s) = 1 | + (0.387 − 0.223i)5-s + (0.285 − 0.494i)7-s + (0.528 + 0.304i)11-s + (0.269 + 0.466i)13-s − 1.18i·17-s + 0.421·19-s + (−0.252 + 0.145i)23-s + (0.100 − 0.173i)25-s + (1.40 + 0.809i)29-s + (−0.467 − 0.810i)31-s − 0.255i·35-s + 0.0540·37-s + (−0.283 + 0.163i)41-s + (0.0813 − 0.140i)43-s + (0.618 + 0.356i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.387278459\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.387278459\) |
\(L(2)\) |
|
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 | \( 1 + (-2 + 3.46i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-5.80 - 3.35i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-3.5 - 6.06i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 20.1iT - 289T^{2} \) |
| 19 | \( 1 - 8T + 361T^{2} \) |
| 23 | \( 1 + (5.80 - 3.35i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-40.6 - 23.4i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (14.5 + 25.1i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 2T + 1.36e3T^{2} \) |
| 41 | \( 1 + (11.6 - 6.70i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-3.5 + 6.06i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-29.0 - 16.7i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 93.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-34.8 + 20.1i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (31 - 53.6i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-29 - 50.2i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 53.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 52T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-24.5 + 42.4i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (11.6 + 6.70i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 107. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-17 + 29.4i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.186702072539971717524786835004, −8.415008729049242554763400180314, −7.39501319655987011218434608850, −6.82618018519194208686893780821, −5.85568604005676540115321117120, −4.90329751072685881754224048485, −4.19438154941279917533481281577, −3.05830996482848064030831375529, −1.84346147574778854551574914282, −0.77611087880107809444988799286,
1.06674401821678764807363768299, 2.19869979302668064084692557742, 3.26217676047341639371303843804, 4.23012246332210650562020163787, 5.34798207824931710943898732225, 6.04180675168862028860488551250, 6.73914166040774284137898112584, 7.84838121364051102454086990325, 8.537541158813949034620552401348, 9.187412932626003273175488787749