L(s) = 1 | + (1.75 − 1.38i)5-s + i·7-s − 5.14·11-s + 4.64i·13-s − 3.86i·17-s + 0.778·19-s + 5.00i·23-s + (1.14 − 4.86i)25-s + 9.42·29-s + 4.72·31-s + (1.38 + 1.75i)35-s + 6i·37-s + 1.00·41-s + 7.00i·43-s + 11.4i·47-s + ⋯ |
L(s) = 1 | + (0.783 − 0.621i)5-s + 0.377i·7-s − 1.55·11-s + 1.28i·13-s − 0.938i·17-s + 0.178·19-s + 1.04i·23-s + (0.228 − 0.973i)25-s + 1.74·29-s + 0.848·31-s + (0.234 + 0.296i)35-s + 0.986i·37-s + 0.157·41-s + 1.06i·43-s + 1.66i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.806989690\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.806989690\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.75 + 1.38i)T \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 + 5.14T + 11T^{2} \) |
| 13 | \( 1 - 4.64iT - 13T^{2} \) |
| 17 | \( 1 + 3.86iT - 17T^{2} \) |
| 19 | \( 1 - 0.778T + 19T^{2} \) |
| 23 | \( 1 - 5.00iT - 23T^{2} \) |
| 29 | \( 1 - 9.42T + 29T^{2} \) |
| 31 | \( 1 - 4.72T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 1.00T + 41T^{2} \) |
| 43 | \( 1 - 7.00iT - 43T^{2} \) |
| 47 | \( 1 - 11.4iT - 47T^{2} \) |
| 53 | \( 1 - 7.55iT - 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 + 11.7iT - 67T^{2} \) |
| 71 | \( 1 + 2.72T + 71T^{2} \) |
| 73 | \( 1 + 5.00iT - 73T^{2} \) |
| 79 | \( 1 + 5.68T + 79T^{2} \) |
| 83 | \( 1 + 4.67iT - 83T^{2} \) |
| 89 | \( 1 + 2.82T + 89T^{2} \) |
| 97 | \( 1 + 1.58iT - 97T^{2} \) |
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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
−9.021980896498859444710872882651, −8.309380910320847275279505237170, −7.55242805150135453154945149033, −6.57133296460824357482357601348, −5.86322420158536554422306084464, −4.92943253575542732332143446993, −4.59994488451215588399093542221, −2.99053629160952327448134668438, −2.31992878004315136132630275793, −1.11373875470825605196781497375,
0.67034462645585601602404197236, 2.27923785208909139433265285932, 2.85139326443831076600829781629, 3.91392252897854704702633376272, 5.16841492202844994201900702739, 5.59231192321348650903620388487, 6.56535488839966997246697963980, 7.22552526780389756924294943919, 8.236343278901774575263465065883, 8.530601551096776172119608860710