L(s) = 1 | + 7.46i·2-s − 39.6·4-s + 35.9i·5-s − 39.8·7-s − 176. i·8-s − 268.·10-s + 188. i·11-s − 80.3·13-s − 297. i·14-s + 681.·16-s − 26.4i·17-s − 465.·19-s − 1.42e3i·20-s − 1.40e3·22-s − 773. i·23-s + ⋯ |
L(s) = 1 | + 1.86i·2-s − 2.47·4-s + 1.43i·5-s − 0.814·7-s − 2.75i·8-s − 2.68·10-s + 1.55i·11-s − 0.475·13-s − 1.51i·14-s + 2.66·16-s − 0.0916i·17-s − 1.29·19-s − 3.56i·20-s − 2.90·22-s − 1.46i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.1929635223\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1929635223\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 59 | \( 1 + 453. iT \) |
good | 2 | \( 1 - 7.46iT - 16T^{2} \) |
| 5 | \( 1 - 35.9iT - 625T^{2} \) |
| 7 | \( 1 + 39.8T + 2.40e3T^{2} \) |
| 11 | \( 1 - 188. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 80.3T + 2.85e4T^{2} \) |
| 17 | \( 1 + 26.4iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 465.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 773. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.23e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 767.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 2.58e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.80e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 2.70e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.55e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 2.98e3iT - 7.89e6T^{2} \) |
| 61 | \( 1 - 3.05e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 8.27e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 609. iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 375.T + 2.83e7T^{2} \) |
| 79 | \( 1 + 6.07e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.21e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 6.07e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 6.24e3T + 8.85e7T^{2} \) |
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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
−10.84110991390667085953118460266, −10.01095779982270047785165679543, −9.346781873146931919025150771561, −8.224165173606183397836089714278, −7.15929828331693726821211467150, −6.83770937451170944063791259596, −6.17892217454372527887357136539, −4.89619742197229122916174288165, −3.94979383254126123127156791410, −2.52437863546902077724988043602,
0.07047028350370047188932703562, 0.77918344268203708926749932959, 1.99350639569630617578909759976, 3.26117478488645599983265409233, 4.08273825354690997002588843196, 5.10376990784243649796770249583, 6.07083776908954407299282014368, 8.120543363885047750730407700227, 8.695640668506754401577110361091, 9.551529178095175494377683941327