L(s) = 1 | + (−5.10e3 − 5.10e3i)2-s + (−7.18e6 + 7.18e6i)3-s − 4.24e9i·4-s + (1.52e11 + 1.16e10i)5-s + 7.32e10·6-s + (2.17e13 + 2.17e13i)7-s + (−4.35e13 + 4.35e13i)8-s + 1.74e15i·9-s + (−7.16e14 − 8.35e14i)10-s − 5.19e16·11-s + (3.04e16 + 3.04e16i)12-s + (6.04e17 − 6.04e17i)13-s − 2.21e17i·14-s + (−1.17e18 + 1.00e18i)15-s − 1.77e19·16-s + (2.55e19 + 2.55e19i)17-s + ⋯ |
L(s) = 1 | + (−0.0778 − 0.0778i)2-s + (−0.166 + 0.166i)3-s − 0.987i·4-s + (0.997 + 0.0764i)5-s + 0.0259·6-s + (0.653 + 0.653i)7-s + (−0.154 + 0.154i)8-s + 0.944i·9-s + (−0.0716 − 0.0835i)10-s − 1.13·11-s + (0.164 + 0.164i)12-s + (0.907 − 0.907i)13-s − 0.101i·14-s + (−0.179 + 0.153i)15-s − 0.963·16-s + (0.524 + 0.524i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{33}{2})\) |
\(\approx\) |
\(2.211175145\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.211175145\) |
\(L(17)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.52e11 - 1.16e10i)T \) |
good | 2 | \( 1 + (5.10e3 + 5.10e3i)T + 4.29e9iT^{2} \) |
| 3 | \( 1 + (7.18e6 - 7.18e6i)T - 1.85e15iT^{2} \) |
| 7 | \( 1 + (-2.17e13 - 2.17e13i)T + 1.10e27iT^{2} \) |
| 11 | \( 1 + 5.19e16T + 2.11e33T^{2} \) |
| 13 | \( 1 + (-6.04e17 + 6.04e17i)T - 4.42e35iT^{2} \) |
| 17 | \( 1 + (-2.55e19 - 2.55e19i)T + 2.36e39iT^{2} \) |
| 19 | \( 1 - 1.62e20iT - 8.31e40T^{2} \) |
| 23 | \( 1 + (2.79e21 - 2.79e21i)T - 3.76e43iT^{2} \) |
| 29 | \( 1 - 3.52e23iT - 6.26e46T^{2} \) |
| 31 | \( 1 - 4.98e23T + 5.29e47T^{2} \) |
| 37 | \( 1 + (-1.32e25 - 1.32e25i)T + 1.52e50iT^{2} \) |
| 41 | \( 1 - 1.17e26T + 4.06e51T^{2} \) |
| 43 | \( 1 + (-9.57e25 + 9.57e25i)T - 1.86e52iT^{2} \) |
| 47 | \( 1 + (4.63e26 + 4.63e26i)T + 3.21e53iT^{2} \) |
| 53 | \( 1 + (7.08e26 - 7.08e26i)T - 1.50e55iT^{2} \) |
| 59 | \( 1 + 1.33e28iT - 4.64e56T^{2} \) |
| 61 | \( 1 - 2.13e28T + 1.35e57T^{2} \) |
| 67 | \( 1 + (-2.29e29 - 2.29e29i)T + 2.71e58iT^{2} \) |
| 71 | \( 1 + 3.37e28T + 1.73e59T^{2} \) |
| 73 | \( 1 + (4.11e29 - 4.11e29i)T - 4.22e59iT^{2} \) |
| 79 | \( 1 + 9.94e29iT - 5.29e60T^{2} \) |
| 83 | \( 1 + (-1.82e30 + 1.82e30i)T - 2.57e61iT^{2} \) |
| 89 | \( 1 - 3.30e30iT - 2.40e62T^{2} \) |
| 97 | \( 1 + (-5.20e31 - 5.20e31i)T + 3.77e63iT^{2} \) |
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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
−15.93429805934349669497120212348, −14.52571914865889949252366044175, −13.19423748486611198607685484744, −10.91780041174010233580945045689, −10.06956426739946515373938191042, −8.248161476369754246647610862910, −5.81406682060566338002063451438, −5.19526684384195570584260212193, −2.43348620295077394769335162215, −1.27122175707047454206767085485,
0.794943427505911133853280563008, 2.53990770475338674832202283236, 4.31159590815782959088594402218, 6.24617210671112776543657319156, 7.78496481260612859198695196432, 9.373170274500095469945295494129, 11.23558031307004616682511784808, 12.84673229761294820169770295740, 14.02577415603583880284745817597, 16.14482400052267070255025609452