L(s) = 1 | + 1.41i·2-s + (2.97 − 0.405i)3-s − 2.00·4-s − 6.32i·5-s + (0.573 + 4.20i)6-s + 5.04·7-s − 2.82i·8-s + (8.67 − 2.41i)9-s + 8.94·10-s + 0.319i·11-s + (−5.94 + 0.811i)12-s + 8.36·13-s + 7.13i·14-s + (−2.56 − 18.7i)15-s + 4.00·16-s + 14.8i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.990 − 0.135i)3-s − 0.500·4-s − 1.26i·5-s + (0.0956 + 0.700i)6-s + 0.720·7-s − 0.353i·8-s + (0.963 − 0.268i)9-s + 0.894·10-s + 0.0290i·11-s + (−0.495 + 0.0676i)12-s + 0.643·13-s + 0.509i·14-s + (−0.171 − 1.25i)15-s + 0.250·16-s + 0.872i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.135i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.990 - 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.94046 + 0.131829i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.94046 + 0.131829i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (-2.97 + 0.405i)T \) |
| 23 | \( 1 - 4.79iT \) |
good | 5 | \( 1 + 6.32iT - 25T^{2} \) |
| 7 | \( 1 - 5.04T + 49T^{2} \) |
| 11 | \( 1 - 0.319iT - 121T^{2} \) |
| 13 | \( 1 - 8.36T + 169T^{2} \) |
| 17 | \( 1 - 14.8iT - 289T^{2} \) |
| 19 | \( 1 + 29.3T + 361T^{2} \) |
| 29 | \( 1 - 27.3iT - 841T^{2} \) |
| 31 | \( 1 + 1.04T + 961T^{2} \) |
| 37 | \( 1 - 10.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 44.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 70.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 76.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 16.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 80.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 92.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 44.6T + 4.48e3T^{2} \) |
| 71 | \( 1 - 71.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 128.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 19.3T + 6.24e3T^{2} \) |
| 83 | \( 1 - 132. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 56.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 58.7T + 9.40e3T^{2} \) |
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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
−13.01331011918687826488931136345, −12.50579312890574912061864242362, −10.79064719265233442529108072428, −9.367465509165465354417995905493, −8.466620638655532573096670584821, −8.081658270645647948289549787710, −6.53703068505629809077320054563, −5.00089781158530690384333475785, −3.94271646465184757052594783232, −1.56992325444721499768121275269,
2.09266185306370401155654514966, 3.26157322485828042430714002250, 4.54335266419359323255392056945, 6.54486837390275178117414626607, 7.83578197276470543250534767930, 8.780219336696952766191326274967, 10.03239434542102765144735282634, 10.79801791036207549021831221938, 11.69288802806402928567450068265, 13.14859600373956952030927287923