| L(s) = 1 | + 1.73i·2-s + 1.00·4-s + 6.78·5-s − 11.7i·7-s + 8.66i·8-s + 11.7i·10-s + (−6.78 + 8.66i)11-s + 11.7i·13-s + 20.3·14-s − 10.9·16-s − 10.3i·17-s + 6.78·20-s + (−15 − 11.7i)22-s − 33.9·23-s + 21·25-s − 20.3·26-s + ⋯ |
| L(s) = 1 | + 0.866i·2-s + 0.250·4-s + 1.35·5-s − 1.67i·7-s + 1.08i·8-s + 1.17i·10-s + (−0.616 + 0.787i)11-s + 0.903i·13-s + 1.45·14-s − 0.687·16-s − 0.611i·17-s + 0.339·20-s + (−0.681 − 0.533i)22-s − 1.47·23-s + 0.839·25-s − 0.782·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.616 - 0.787i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.616 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.54050 + 0.750246i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.54050 + 0.750246i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (6.78 - 8.66i)T \) |
| good | 2 | \( 1 - 1.73iT - 4T^{2} \) |
| 5 | \( 1 - 6.78T + 25T^{2} \) |
| 7 | \( 1 + 11.7iT - 49T^{2} \) |
| 13 | \( 1 - 11.7iT - 169T^{2} \) |
| 17 | \( 1 + 10.3iT - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 + 33.9T + 529T^{2} \) |
| 29 | \( 1 + 34.6iT - 841T^{2} \) |
| 31 | \( 1 + 10T + 961T^{2} \) |
| 37 | \( 1 - 50T + 1.36e3T^{2} \) |
| 41 | \( 1 - 34.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 46.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 33.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 33.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 67.8T + 3.48e3T^{2} \) |
| 61 | \( 1 - 58.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 10T + 4.48e3T^{2} \) |
| 71 | \( 1 + 33.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + 70.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 58.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 76.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 13.5T + 7.92e3T^{2} \) |
| 97 | \( 1 + 40T + 9.40e3T^{2} \) |
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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
−13.89922072341241488410290427429, −13.26871859530544128937147867836, −11.59111234752925085496808151154, −10.34555912405217363352062594668, −9.634373405104048947142710018896, −7.86181818641162266416338414134, −6.96869285574314216953648404092, −5.98254276374927009077012157907, −4.53034004993253038586929264213, −2.07236184707322903606294201756,
1.96821590922520266313627769244, 2.99748397720649820475135173984, 5.55242630285696233660934242967, 6.18055188754524765726142955172, 8.243148484179698270246004268640, 9.459031267723873954939640785931, 10.31981629847337504815215965401, 11.34858889270193298032883116351, 12.54463015180182815196002007968, 13.07267612789933894968377726030
