L(s) = 1 | + (−2.57 + 2.57i)5-s − 3.74·7-s + (−3.64 − 3.64i)11-s + (−4.64 + 4.64i)13-s + 0.913i·17-s + (1.41 + 1.41i)19-s − 6i·23-s − 8.29i·25-s + (1.66 + 1.66i)29-s + 6.57i·31-s + (9.64 − 9.64i)35-s + (−1 − i)37-s + 0.913·41-s + (−3.74 + 3.74i)43-s + 6·47-s + ⋯ |
L(s) = 1 | + (−1.15 + 1.15i)5-s − 1.41·7-s + (−1.09 − 1.09i)11-s + (−1.28 + 1.28i)13-s + 0.221i·17-s + (0.324 + 0.324i)19-s − 1.25i·23-s − 1.65i·25-s + (0.309 + 0.309i)29-s + 1.18i·31-s + (1.63 − 1.63i)35-s + (−0.164 − 0.164i)37-s + 0.142·41-s + (−0.570 + 0.570i)43-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3694261173\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3694261173\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.57 - 2.57i)T - 5iT^{2} \) |
| 7 | \( 1 + 3.74T + 7T^{2} \) |
| 11 | \( 1 + (3.64 + 3.64i)T + 11iT^{2} \) |
| 13 | \( 1 + (4.64 - 4.64i)T - 13iT^{2} \) |
| 17 | \( 1 - 0.913iT - 17T^{2} \) |
| 19 | \( 1 + (-1.41 - 1.41i)T + 19iT^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + (-1.66 - 1.66i)T + 29iT^{2} \) |
| 31 | \( 1 - 6.57iT - 31T^{2} \) |
| 37 | \( 1 + (1 + i)T + 37iT^{2} \) |
| 41 | \( 1 - 0.913T + 41T^{2} \) |
| 43 | \( 1 + (3.74 - 3.74i)T - 43iT^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + (-3.49 + 3.49i)T - 53iT^{2} \) |
| 59 | \( 1 + (-1.29 - 1.29i)T + 59iT^{2} \) |
| 61 | \( 1 + (-0.291 + 0.291i)T - 61iT^{2} \) |
| 67 | \( 1 + (-3.32 - 3.32i)T + 67iT^{2} \) |
| 71 | \( 1 + 13.2iT - 71T^{2} \) |
| 73 | \( 1 - 8.58iT - 73T^{2} \) |
| 79 | \( 1 + 8.39iT - 79T^{2} \) |
| 83 | \( 1 + (4.93 - 4.93i)T - 83iT^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−8.859764614991955158936718908753, −8.108549040545855775410316896708, −7.18321867502607391145185470663, −6.81129829417444715831079121429, −6.03517882452663449896923968587, −4.86206950823951594132021751250, −3.84233972152559760343141083171, −3.08751342036721097181182341929, −2.52609498634542161018983091134, −0.22629666858793277242949230111,
0.59208385718503861023620850191, 2.45517984129134354210366263970, 3.31608915378822602859906485225, 4.29674323988908085992860367769, 5.10567611312266549203885440377, 5.68171710457062661806328642016, 7.12898066283848357918233475591, 7.51300898152284225319511153874, 8.149829860213619428521021926327, 9.161333755176295629734191846233