| L(s) = 1 | − 2.44i·5-s + 2i·7-s − 4.89·11-s + 3.46·13-s + 4.24i·17-s − 6.92i·19-s − 8.48·23-s − 0.999·25-s + 7.34i·29-s + 2i·31-s + 4.89·35-s − 6.92·37-s − 4.24i·41-s − 8.48·47-s + 3·49-s + ⋯ |
| L(s) = 1 | − 1.09i·5-s + 0.755i·7-s − 1.47·11-s + 0.960·13-s + 1.02i·17-s − 1.58i·19-s − 1.76·23-s − 0.199·25-s + 1.36i·29-s + 0.359i·31-s + 0.828·35-s − 1.13·37-s − 0.662i·41-s − 1.23·47-s + 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2232434287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2232434287\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.44iT - 5T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 - 4.24iT - 17T^{2} \) |
| 19 | \( 1 + 6.92iT - 19T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 - 7.34iT - 29T^{2} \) |
| 31 | \( 1 - 2iT - 31T^{2} \) |
| 37 | \( 1 + 6.92T + 37T^{2} \) |
| 41 | \( 1 + 4.24iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 + 2.44iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 6.92T + 61T^{2} \) |
| 67 | \( 1 - 6.92iT - 67T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 14iT - 79T^{2} \) |
| 83 | \( 1 + 4.89T + 83T^{2} \) |
| 89 | \( 1 - 4.24iT - 89T^{2} \) |
| 97 | \( 1 + 16T + 97T^{2} \) |
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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
−8.980617214494359811795089196111, −8.614787714973299984385257347702, −8.075665247862977718548993572839, −7.01701784866221970354286443877, −5.98392146223115976599147993358, −5.34165462412226291459088963407, −4.71774505049528075411528165551, −3.62594597739399753820412261891, −2.51771305692890484354533740947, −1.47217650245451271070185182662,
0.07248829318256499730257457055, 1.82077966882060344476605585218, 2.90817296788630084548958929099, 3.66327300233158772961756106065, 4.58048406592045196818958946687, 5.79543714301236877978775245509, 6.25041419373697848409490294941, 7.34582246207506604954450609908, 7.78000489734727353562321592439, 8.460288366988288650961601308314
