L(s) = 1 | − i·2-s + 3-s − 4-s − i·5-s − i·6-s + 2.82i·7-s + i·8-s + 9-s − 10-s − 5.65i·11-s − 12-s + 2.82·14-s − i·15-s + 16-s + 4.82·17-s − i·18-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.447i·5-s − 0.408i·6-s + 1.06i·7-s + 0.353i·8-s + 0.333·9-s − 0.316·10-s − 1.70i·11-s − 0.288·12-s + 0.755·14-s − 0.258i·15-s + 0.250·16-s + 1.17·17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.670859476\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.670859476\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 - 2.82iT - 19T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 + 3.17T + 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + 0.343iT - 37T^{2} \) |
| 41 | \( 1 + 3.65iT - 41T^{2} \) |
| 43 | \( 1 - 1.65T + 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 9.31T + 53T^{2} \) |
| 59 | \( 1 + 13.6iT - 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 5.65iT - 67T^{2} \) |
| 71 | \( 1 + 5.65iT - 71T^{2} \) |
| 73 | \( 1 - 2.48iT - 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 17.6iT - 83T^{2} \) |
| 89 | \( 1 + 4.34iT - 89T^{2} \) |
| 97 | \( 1 - 8.82iT - 97T^{2} \) |
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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
−8.263084503358621180232143749834, −7.61191645757584232675945905518, −6.11829013591855079967526129718, −5.78705157754678762448826726873, −5.03224949295865613549263309949, −3.76449737491834952027846304472, −3.47858023847741537526223747609, −2.43028423342152837668000240245, −1.67748287910848453309234054537, −0.41943137717882512328225123036,
1.25154585277326469834561750855, 2.30945374016526034807096000167, 3.41429172174849349860827423339, 4.15894528739042375936107893188, 4.71006263232220458835281489748, 5.71396093458859458436070845236, 6.64632964497744372176301979755, 7.18226823825778772993151905846, 7.72370367308387830953466544204, 8.152881862537053266580157604145