| L(s) = 1 | + (0.0440 + 0.0254i)5-s + (4.52 + 7.84i)7-s + (−3.29 + 1.90i)11-s + (0.216 − 0.375i)13-s + 26.2i·17-s + 34.2·19-s + (29.9 + 17.3i)23-s + (−12.4 − 21.6i)25-s + (−14.0 + 8.10i)29-s + (−17.1 + 29.7i)31-s + 0.460i·35-s + 29.2·37-s + (−48.7 − 28.1i)41-s + (−3.94 − 6.83i)43-s + (33.4 − 19.3i)47-s + ⋯ |
| L(s) = 1 | + (0.00880 + 0.00508i)5-s + (0.647 + 1.12i)7-s + (−0.299 + 0.172i)11-s + (0.0166 − 0.0288i)13-s + 1.54i·17-s + 1.80·19-s + (1.30 + 0.752i)23-s + (−0.499 − 0.865i)25-s + (−0.483 + 0.279i)29-s + (−0.553 + 0.959i)31-s + 0.0131i·35-s + 0.791·37-s + (−1.18 − 0.685i)41-s + (−0.0917 − 0.158i)43-s + (0.711 − 0.410i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.42372 + 0.708015i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42372 + 0.708015i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-0.0440 - 0.0254i)T + (12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (-4.52 - 7.84i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (3.29 - 1.90i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-0.216 + 0.375i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 26.2iT - 289T^{2} \) |
| 19 | \( 1 - 34.2T + 361T^{2} \) |
| 23 | \( 1 + (-29.9 - 17.3i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (14.0 - 8.10i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (17.1 - 29.7i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 29.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + (48.7 + 28.1i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (3.94 + 6.83i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-33.4 + 19.3i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 50.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-8.54 - 4.93i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (36.5 + 63.3i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (12.6 - 21.9i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 97.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 77.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + (42.1 + 72.9i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (40.6 - 23.4i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 108. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (32.4 + 56.1i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−12.15401910367854979681288553500, −11.39184342290847615382811956448, −10.32519573756307969531890030866, −9.181570987627718843877859403468, −8.330918558305479213339906044328, −7.28128768998186305071061415380, −5.81545952324260937857258169720, −5.02703927177714132515297135633, −3.33960968152408349756510724051, −1.76429894830359508876465030460,
0.989399795376746464926906543620, 3.02473180882254094199143850690, 4.49341275933664568170475580854, 5.51411593097828802737726511448, 7.21337823628303826398339352794, 7.62533938782479897555235922883, 9.103051895612425925461105955159, 9.994016370970845996116237757524, 11.16892017253011616202388744033, 11.64166653462482130015372612010
