L(s) = 1 | + (1 + i)2-s + (1.22 − 1.22i)3-s + 2i·4-s + (4.99 − 0.177i)5-s + 2.44·6-s + (−4.63 − 4.63i)7-s + (−2 + 2i)8-s − 2.99i·9-s + (5.17 + 4.81i)10-s + 2.27·11-s + (2.44 + 2.44i)12-s + (6.59 − 6.59i)13-s − 9.27i·14-s + (5.90 − 6.33i)15-s − 4·16-s + (16.0 + 16.0i)17-s + ⋯ |
L(s) = 1 | + (0.5 + 0.5i)2-s + (0.408 − 0.408i)3-s + 0.5i·4-s + (0.999 − 0.0354i)5-s + 0.408·6-s + (−0.662 − 0.662i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + (0.517 + 0.481i)10-s + 0.206·11-s + (0.204 + 0.204i)12-s + (0.507 − 0.507i)13-s − 0.662i·14-s + (0.393 − 0.422i)15-s − 0.250·16-s + (0.946 + 0.946i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.215309417\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.215309417\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - i)T \) |
| 3 | \( 1 + (-1.22 + 1.22i)T \) |
| 5 | \( 1 + (-4.99 + 0.177i)T \) |
| 23 | \( 1 + (-3.39 + 3.39i)T \) |
good | 7 | \( 1 + (4.63 + 4.63i)T + 49iT^{2} \) |
| 11 | \( 1 - 2.27T + 121T^{2} \) |
| 13 | \( 1 + (-6.59 + 6.59i)T - 169iT^{2} \) |
| 17 | \( 1 + (-16.0 - 16.0i)T + 289iT^{2} \) |
| 19 | \( 1 + 33.4iT - 361T^{2} \) |
| 29 | \( 1 - 8.75iT - 841T^{2} \) |
| 31 | \( 1 - 26.0T + 961T^{2} \) |
| 37 | \( 1 + (-18.6 - 18.6i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 4.34T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-33.4 + 33.4i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (28.7 + 28.7i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (34.3 - 34.3i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 71.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 71.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + (80.0 + 80.0i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 103.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-1.17 + 1.17i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 134. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (11.8 - 11.8i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 40.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-31.1 - 31.1i)T + 9.40e3iT^{2} \) |
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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
−10.16922782751947814790944404747, −9.272656181730805956169392031419, −8.465198957807357226574556520943, −7.42677330632621251860810600694, −6.57891483769406240926013181025, −5.98245446455459346354356537693, −4.84362721774153066325107749072, −3.57582703417283138532783384564, −2.64491731350311212661244582944, −1.04698004881103192869440232366,
1.47659091036342258747391367894, 2.67686847922439919852120460735, 3.51057406021579323694701722616, 4.74832109296280423639256281457, 5.83722580404574220492050288683, 6.30178147382129072853317554471, 7.75931780162269944226792739208, 8.952935963129503576641696907777, 9.642141650357431253503057654288, 10.05165455964656339527638996129