| L(s) = 1 | − 3.46i·5-s + 3.46i·7-s + 3·11-s − 4·13-s + 1.73i·17-s − 1.73i·19-s − 6.99·25-s + 3.46i·29-s + 11.9·35-s − 2·37-s + 5.19i·41-s + 5.19i·43-s − 12·47-s − 4.99·49-s − 10.3i·55-s + ⋯ |
| L(s) = 1 | − 1.54i·5-s + 1.30i·7-s + 0.904·11-s − 1.10·13-s + 0.420i·17-s − 0.397i·19-s − 1.39·25-s + 0.643i·29-s + 2.02·35-s − 0.328·37-s + 0.811i·41-s + 0.792i·43-s − 1.75·47-s − 0.714·49-s − 1.40i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.048677897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.048677897\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 1.73iT - 17T^{2} \) |
| 19 | \( 1 + 1.73iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 3.46iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 5.19iT - 41T^{2} \) |
| 43 | \( 1 - 5.19iT - 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 15T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 8.66iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 + 3.46iT - 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 13.8iT - 89T^{2} \) |
| 97 | \( 1 - 13T + 97T^{2} \) |
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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
−8.544146853427248120847010298072, −7.909104865861819619266446913079, −6.89657598497463148020203748383, −6.12117267776560531834819161570, −5.31104750084759759343467302559, −4.88696217949445653420382019763, −4.12315867435088967987816226232, −2.98857828378875688367914182606, −2.00511844363563346756604057847, −1.15861876444581768248821836592,
0.28535962022084670434016196853, 1.72530005874072464053233690917, 2.71389588890699530735884162776, 3.54946211292844372993121295468, 4.10920124358460771793855052000, 5.04639152747543618677694527593, 6.13131401129052213773272265564, 6.77458589464817829041759209172, 7.25341786170802775526048816972, 7.67585769261164859954415447825
