L(s) = 1 | + 7.77i·2-s − 44.4·4-s − 28.5·5-s + 72.7·7-s − 221. i·8-s − 222. i·10-s + 9.33i·11-s − 106. i·13-s + 565. i·14-s + 1.01e3·16-s − 337.·17-s + 546.·19-s + 1.27e3·20-s − 72.6·22-s − 39.3i·23-s + ⋯ |
L(s) = 1 | + 1.94i·2-s − 2.78·4-s − 1.14·5-s + 1.48·7-s − 3.46i·8-s − 2.22i·10-s + 0.0771i·11-s − 0.631i·13-s + 2.88i·14-s + 3.95·16-s − 1.16·17-s + 1.51·19-s + 3.17·20-s − 0.150·22-s − 0.0742i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 + 0.506i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.861 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.108729046\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.108729046\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 59 | \( 1 + (-3.00e3 + 1.76e3i)T \) |
good | 2 | \( 1 - 7.77iT - 16T^{2} \) |
| 5 | \( 1 + 28.5T + 625T^{2} \) |
| 7 | \( 1 - 72.7T + 2.40e3T^{2} \) |
| 11 | \( 1 - 9.33iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 106. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 337.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 546.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 39.3iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 444.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.23e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.94e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 835.T + 2.82e6T^{2} \) |
| 43 | \( 1 + 2.24e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 3.91e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.82e3T + 7.89e6T^{2} \) |
| 61 | \( 1 + 2.34e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 3.65e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 2.63e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 8.92e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 1.49e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 3.98e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 2.53e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 9.21e3iT - 8.85e7T^{2} \) |
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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
−10.68231242669486584792733299541, −9.379528499663163692840103977719, −8.443680110782027858538692055932, −7.922779128205248318095135081464, −7.36920760424655264062317015497, −6.36240561372821936593293232407, −5.01204835220778833259710217552, −4.75029656159066041905775402934, −3.53489346001145562121005219565, −0.963506264450913973378672275150,
0.39283917695136928030351070182, 1.52222877039916514033468849736, 2.57841486964444615601978857545, 3.91663432393340475969410896776, 4.41878715811062747372776216900, 5.36416213147957076957711061113, 7.47228863771794987282159369868, 8.259139700596342685430606394766, 8.996252106611364249981577748326, 9.939018934199817710525221119912