L(s) = 1 | + 1.41·2-s + 2.00·4-s − 1.43i·5-s + 2.82·8-s − 2.02i·10-s − 6·11-s + 21.3i·13-s + 4.00·16-s + 8.95i·17-s + 7.22i·19-s − 2.86i·20-s − 8.48·22-s + 37.4·23-s + 22.9·25-s + 30.1i·26-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.500·4-s − 0.286i·5-s + 0.353·8-s − 0.202i·10-s − 0.545·11-s + 1.64i·13-s + 0.250·16-s + 0.526i·17-s + 0.380i·19-s − 0.143i·20-s − 0.385·22-s + 1.62·23-s + 0.917·25-s + 1.16i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.876097927\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.876097927\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 1.43iT - 25T^{2} \) |
| 11 | \( 1 + 6T + 121T^{2} \) |
| 13 | \( 1 - 21.3iT - 169T^{2} \) |
| 17 | \( 1 - 8.95iT - 289T^{2} \) |
| 19 | \( 1 - 7.22iT - 361T^{2} \) |
| 23 | \( 1 - 37.4T + 529T^{2} \) |
| 29 | \( 1 - 33.9T + 841T^{2} \) |
| 31 | \( 1 - 44.1iT - 961T^{2} \) |
| 37 | \( 1 + 27.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 54.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.48T + 1.84e3T^{2} \) |
| 47 | \( 1 + 43.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 85.4T + 2.80e3T^{2} \) |
| 59 | \( 1 - 41.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 1.18iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.39T + 4.48e3T^{2} \) |
| 71 | \( 1 + 137.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 78.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 98.3T + 6.24e3T^{2} \) |
| 83 | \( 1 - 110. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 20.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 10.9iT - 9.40e3T^{2} \) |
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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
−10.29473466582008074490111425266, −8.990553908895554637204603509944, −8.527237602301258726966313488172, −7.10553486110165627117568790482, −6.72187296788332888169177156418, −5.44478535288247550985165008458, −4.74999101737777855099276256939, −3.77778368017826621547247467668, −2.60748244694659129057416660886, −1.34402907799877051047905136245,
0.813772070787579504782818704325, 2.68718120906911471775970047035, 3.17229138258938035708769224771, 4.66196562839233124967753877586, 5.30423648262909056679696322236, 6.29202151900418024583195404286, 7.23172214448776729283519923480, 7.962093504669229668367406607489, 8.971703248227301080408824487833, 10.12415726731600549065158915064