| L(s) = 1 | − 1.23·3-s − 1.23·7-s − 7.47·9-s − 11.4i·11-s + 5.41i·13-s + 6.94i·17-s + 29.8i·19-s + 1.52·21-s − 19.1·23-s + 20.3·27-s + 21.0·29-s − 34.4i·31-s + 14.1i·33-s + 19.3i·37-s − 6.69i·39-s + ⋯ |
| L(s) = 1 | − 0.412·3-s − 0.176·7-s − 0.830·9-s − 1.03i·11-s + 0.416i·13-s + 0.408i·17-s + 1.57i·19-s + 0.0727·21-s − 0.831·23-s + 0.754·27-s + 0.726·29-s − 1.11i·31-s + 0.427i·33-s + 0.521i·37-s − 0.171i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.198776682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.198776682\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 1.23T + 9T^{2} \) |
| 7 | \( 1 + 1.23T + 49T^{2} \) |
| 11 | \( 1 + 11.4iT - 121T^{2} \) |
| 13 | \( 1 - 5.41iT - 169T^{2} \) |
| 17 | \( 1 - 6.94iT - 289T^{2} \) |
| 19 | \( 1 - 29.8iT - 361T^{2} \) |
| 23 | \( 1 + 19.1T + 529T^{2} \) |
| 29 | \( 1 - 21.0T + 841T^{2} \) |
| 31 | \( 1 + 34.4iT - 961T^{2} \) |
| 37 | \( 1 - 19.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 58.1T + 1.68e3T^{2} \) |
| 43 | \( 1 - 62.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 63.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 98.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 19.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 1.19T + 3.72e3T^{2} \) |
| 67 | \( 1 + 5.01T + 4.48e3T^{2} \) |
| 71 | \( 1 + 84.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 70.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 124. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 160.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 46.2T + 7.92e3T^{2} \) |
| 97 | \( 1 - 133. iT - 9.40e3T^{2} \) |
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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
−9.077522397010463808646606125431, −8.329472820513233744996073844781, −7.75902956664287096487765910158, −6.37225847944130294593318622657, −6.06399877904204699009248047514, −5.21154544864792049382269412925, −4.00943951294079600557191547883, −3.21662357916095698174102579532, −1.97206545129857386438288407234, −0.52838539803309648713096520025,
0.70603066617163850220421561828, 2.28358698130821976261609565578, 3.13561848337588660235163979297, 4.45647980826223614027047620288, 5.12818661231357346923237753310, 6.02507143807480002028913603933, 6.87473176358681054061930819011, 7.56266221342236914344148368372, 8.614789918512523294339017453126, 9.219116948771262448383064365428
