L(s) = 1 | − 4.64i·2-s − 5.59·4-s + 0.691·5-s − 76.1·7-s − 48.3i·8-s − 3.21i·10-s − 74.8i·11-s − 105. i·13-s + 353. i·14-s − 314.·16-s + 141.·17-s − 170.·19-s − 3.87·20-s − 347.·22-s − 226. i·23-s + ⋯ |
L(s) = 1 | − 1.16i·2-s − 0.349·4-s + 0.0276·5-s − 1.55·7-s − 0.755i·8-s − 0.0321i·10-s − 0.618i·11-s − 0.623i·13-s + 1.80i·14-s − 1.22·16-s + 0.491·17-s − 0.473·19-s − 0.00967·20-s − 0.718·22-s − 0.428i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.1123545481\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1123545481\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 59 | \( 1 + (2.33e3 - 2.58e3i)T \) |
good | 2 | \( 1 + 4.64iT - 16T^{2} \) |
| 5 | \( 1 - 0.691T + 625T^{2} \) |
| 7 | \( 1 + 76.1T + 2.40e3T^{2} \) |
| 11 | \( 1 + 74.8iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 105. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 141.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 170.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 226. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 677.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 114. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 488. iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.82e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 527. iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 1.90e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 4.68e3T + 7.89e6T^{2} \) |
| 61 | \( 1 - 6.41e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 1.77e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 170.T + 2.54e7T^{2} \) |
| 73 | \( 1 - 8.99e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 204.T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.39e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 3.00e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 9.22e3iT - 8.85e7T^{2} \) |
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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
−10.32758319884334692191419113476, −9.799089414351025663851682463457, −8.953338156012618844468059151491, −7.72007471867307508180995155827, −6.52692963867363625740995895547, −5.89062565175119803562748396457, −4.23010028341207289042705608175, −3.23744823888220944127853057295, −2.62379479601978052633880517104, −1.03847707071242148978072173604,
0.03117198695788596035949288951, 2.06954450938057441156449136407, 3.38485459274211359141912911886, 4.64400304553638932165011727281, 5.86603113488134300279654027707, 6.48155224745483948980415429465, 7.21444446944273092387329164024, 8.101835582557694223558836331395, 9.246920760095729921887482515271, 9.796173896623718745585681364378