| L(s) = 1 | + 0.112i·2-s − 1.73·3-s + 3.98·4-s − 0.195i·6-s + (−1.98 + 6.71i)7-s + 0.902i·8-s + 2.99·9-s − 15.8·11-s − 6.90·12-s − 13.3·13-s + (−0.758 − 0.224i)14-s + 15.8·16-s − 15.6·17-s + 0.338i·18-s − 30.8i·19-s + ⋯ |
| L(s) = 1 | + 0.0564i·2-s − 0.577·3-s + 0.996·4-s − 0.0326i·6-s + (−0.283 + 0.959i)7-s + 0.112i·8-s + 0.333·9-s − 1.44·11-s − 0.575·12-s − 1.02·13-s + (−0.0541 − 0.0160i)14-s + 0.990·16-s − 0.922·17-s + 0.0188i·18-s − 1.62i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.175i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1060379424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1060379424\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 1.73T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.98 - 6.71i)T \) |
| good | 2 | \( 1 - 0.112iT - 4T^{2} \) |
| 11 | \( 1 + 15.8T + 121T^{2} \) |
| 13 | \( 1 + 13.3T + 169T^{2} \) |
| 17 | \( 1 + 15.6T + 289T^{2} \) |
| 19 | \( 1 + 30.8iT - 361T^{2} \) |
| 23 | \( 1 - 3.63iT - 529T^{2} \) |
| 29 | \( 1 + 14.5T + 841T^{2} \) |
| 31 | \( 1 - 11.3iT - 961T^{2} \) |
| 37 | \( 1 + 17.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 27.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 12.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 80.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 55.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 79.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 94.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 103. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 113.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 20.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 1.27T + 6.24e3T^{2} \) |
| 83 | \( 1 + 19.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 131. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 12.4T + 9.40e3T^{2} \) |
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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
−11.24854909669541642831852116260, −10.37886027376293989023411987549, −9.497881883857520977233444269156, −8.350432841114537064221347537972, −7.29233748566847499039100932445, −6.61466673886502222870573805183, −5.52897046951758134826060938215, −4.86339080170259102152042732807, −2.90831154711591665415393234654, −2.18455016827818613541898925201,
0.03884431396000700837653556822, 1.84816440837497974141785929775, 3.12902521298208286815347798714, 4.48624535472445511086057888803, 5.61473835591087665676078671833, 6.54304836712025032295387248020, 7.44763153077729325270386351757, 7.988750098398114805174831758749, 9.718787015168406828568994027998, 10.40171346259733762970968871026
