L(s) = 1 | − 1.73i·3-s − 4.37·5-s − 2.64i·7-s − 2.99·9-s + 3.58i·11-s + 7.58i·15-s + 2.55·17-s + 5.29i·19-s − 4.58·21-s − 0.417i·23-s + 14.1·25-s + 5.19i·27-s + 3.46i·31-s + 6.20·33-s + 11.5i·35-s + ⋯ |
L(s) = 1 | − 0.999i·3-s − 1.95·5-s − 0.999i·7-s − 0.999·9-s + 1.08i·11-s + 1.95i·15-s + 0.618·17-s + 1.21i·19-s − 0.999·21-s − 0.0870i·23-s + 2.83·25-s + 0.999i·27-s + 0.622i·31-s + 1.08·33-s + 1.95i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.179757 + 0.179757i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.179757 + 0.179757i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 + 2.64iT \) |
good | 5 | \( 1 + 4.37T + 5T^{2} \) |
| 11 | \( 1 - 3.58iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 2.55T + 17T^{2} \) |
| 19 | \( 1 - 5.29iT - 19T^{2} \) |
| 23 | \( 1 + 0.417iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 9.16T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 15.5iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−10.85962732570813221922468515207, −10.05125756585521375404398772256, −8.517578888143290049481802800882, −7.983317332814510072091986762286, −7.19588478557558876832686750409, −6.82899403181272537282515344857, −5.16381820817204709891074614840, −4.04672656385123476953049451924, −3.26442526198933076974934335948, −1.39258147639189917845121122488,
0.14498208551501553164315421097, 2.99495342933142430903112229640, 3.57903812301968250769548692205, 4.69258226236986584306249530194, 5.49124495207621961466249516530, 6.78230976098533023421951905511, 8.006156951885215234387563261087, 8.558315782450451290799452317912, 9.203298884185361644849973968202, 10.49046179488580187142184946127