L(s) = 1 | − i·3-s − i·5-s − 3.74·7-s − 9-s + 0.951·11-s − 0.958·13-s − 15-s − 7.12i·17-s − 8.14·19-s + 3.74i·21-s + (3.34 − 3.43i)23-s − 25-s + i·27-s + 3.97·29-s − 10.2i·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.447i·5-s − 1.41·7-s − 0.333·9-s + 0.286·11-s − 0.265·13-s − 0.258·15-s − 1.72i·17-s − 1.86·19-s + 0.818i·21-s + (0.697 − 0.716i)23-s − 0.200·25-s + 0.192i·27-s + 0.738·29-s − 1.84i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1841674883\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1841674883\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (-3.34 + 3.43i)T \) |
good | 7 | \( 1 + 3.74T + 7T^{2} \) |
| 11 | \( 1 - 0.951T + 11T^{2} \) |
| 13 | \( 1 + 0.958T + 13T^{2} \) |
| 17 | \( 1 + 7.12iT - 17T^{2} \) |
| 19 | \( 1 + 8.14T + 19T^{2} \) |
| 29 | \( 1 - 3.97T + 29T^{2} \) |
| 31 | \( 1 + 10.2iT - 31T^{2} \) |
| 37 | \( 1 + 7.97iT - 37T^{2} \) |
| 41 | \( 1 - 2.05T + 41T^{2} \) |
| 43 | \( 1 - 4.47T + 43T^{2} \) |
| 47 | \( 1 - 5.73iT - 47T^{2} \) |
| 53 | \( 1 - 5.84iT - 53T^{2} \) |
| 59 | \( 1 - 5.24iT - 59T^{2} \) |
| 61 | \( 1 - 1.55iT - 61T^{2} \) |
| 67 | \( 1 + 8.31T + 67T^{2} \) |
| 71 | \( 1 - 14.9iT - 71T^{2} \) |
| 73 | \( 1 - 2.66T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 - 3.86T + 83T^{2} \) |
| 89 | \( 1 + 3.08iT - 89T^{2} \) |
| 97 | \( 1 + 5.79iT - 97T^{2} \) |
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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
−7.45614678064388608904870433512, −6.97887920563812808238913337597, −6.22667678988638572632419432262, −5.75076402311436873191600802579, −4.59645564781851457575602369420, −4.02947940886559851186192022925, −2.76322126942181254514241247416, −2.42687420021126207148708566332, −0.867385685733382361409051102809, −0.05854945948714294023653172970,
1.64824236203600840153744239470, 2.80502801055669301654084130976, 3.45408667026583160072483841478, 4.09301622369444806412025894866, 4.97383712472306811075314021200, 6.01465969513600981625198541863, 6.48988734618097643952123094422, 6.94636173425583588035179915036, 8.129220338231806630047037860346, 8.703485770548908919861219920415