L(s) = 1 | − 9i·3-s − 84.4·5-s + 98.5·7-s − 81·9-s + (−362. − 171. i)11-s + 28.0i·13-s + 760. i·15-s − 334. i·17-s − 2.60e3·19-s − 887. i·21-s + 1.42e3i·23-s + 4.01e3·25-s + 729i·27-s − 2.77e3i·29-s + 4.56e3i·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 1.51·5-s + 0.760·7-s − 0.333·9-s + (−0.903 − 0.427i)11-s + 0.0460i·13-s + 0.872i·15-s − 0.280i·17-s − 1.65·19-s − 0.439i·21-s + 0.563i·23-s + 1.28·25-s + 0.192i·27-s − 0.611i·29-s + 0.853i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0812i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.8112323997\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8112323997\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 9iT \) |
| 11 | \( 1 + (362. + 171. i)T \) |
good | 5 | \( 1 + 84.4T + 3.12e3T^{2} \) |
| 7 | \( 1 - 98.5T + 1.68e4T^{2} \) |
| 13 | \( 1 - 28.0iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 334. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.60e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.42e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 2.77e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 4.56e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 5.08e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.62e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 8.78e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 621. iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 8.70e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.33e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 4.35e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 2.01e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 2.36e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 7.79e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 3.75e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.53e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.83e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.51e5T + 8.58e9T^{2} \) |
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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
−10.35423991300917832164389368770, −8.739796772729584112399524824685, −8.190447173191458055989314315531, −7.55869129143439112436008286183, −6.63791463151393546138178868711, −5.32076596724066622555513627553, −4.36335389848929347243035095119, −3.29683850269881306810276563839, −2.01562729754575243750662627161, −0.55582954798108947620197214524,
0.33729434212608270614960895238, 2.12917867951232127981762195893, 3.47273437655374853306447497994, 4.41531458683480692101849101034, 4.99664997560473155177671829222, 6.44943237064832536286861317865, 7.66880512618157409195207567076, 8.164921880881741064590407440471, 8.965592816357790317348223570076, 10.38543119650045634820638572170