L(s) = 1 | + 6.27·5-s − 12.2·7-s + 0.274·11-s − 13.0i·13-s − 17.3·17-s + (−7.54 − 17.4i)19-s + 20.5·23-s + 14.3·25-s − 26.0i·29-s + 23.5i·31-s − 77.0·35-s + 66.7i·37-s − 3.57i·41-s + 48.1·43-s − 12.4·47-s + ⋯ |
L(s) = 1 | + 1.25·5-s − 1.75·7-s + 0.0249·11-s − 1.00i·13-s − 1.02·17-s + (−0.397 − 0.917i)19-s + 0.893·23-s + 0.574·25-s − 0.897i·29-s + 0.758i·31-s − 2.20·35-s + 1.80i·37-s − 0.0872i·41-s + 1.11·43-s − 0.265·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7719024456\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7719024456\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (7.54 + 17.4i)T \) |
good | 5 | \( 1 - 6.27T + 25T^{2} \) |
| 7 | \( 1 + 12.2T + 49T^{2} \) |
| 11 | \( 1 - 0.274T + 121T^{2} \) |
| 13 | \( 1 + 13.0iT - 169T^{2} \) |
| 17 | \( 1 + 17.3T + 289T^{2} \) |
| 23 | \( 1 - 20.5T + 529T^{2} \) |
| 29 | \( 1 + 26.0iT - 841T^{2} \) |
| 31 | \( 1 - 23.5iT - 961T^{2} \) |
| 37 | \( 1 - 66.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 3.57iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 48.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 12.4T + 2.20e3T^{2} \) |
| 53 | \( 1 - 25.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 0.230iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 28.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 102. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 107. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 11.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 26.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 89.6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 139. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 41.3iT - 9.40e3T^{2} \) |
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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
−9.125326105947246695753981496272, −8.307295892198083752440676763193, −7.02183670722568309123740986550, −6.58707992659125035735113752733, −5.92789636946246042892829236897, −5.18825598884575102246588242179, −4.10094313275204776243367873170, −2.87409750681423334031966482645, −2.57558504083076969081856735867, −1.04285223906952864520779138649,
0.18282836705245276394543868731, 1.73777355508200041860713602127, 2.52093924931143569252709519568, 3.51270885425679627595765269591, 4.39425034042689366766813849446, 5.56790425098377351932582906927, 6.20779394164882869193752200527, 6.66048163071282438614873794664, 7.42505602836958960913028675598, 8.799630458287684821505955686823