| L(s) = 1 | − 1.41·2-s + (−2.88 − 0.812i)3-s + 2.00·4-s + (4.08 + 1.14i)6-s + 2.64i·7-s − 2.82·8-s + (7.68 + 4.69i)9-s − 19.7i·11-s + (−5.77 − 1.62i)12-s + 18.5i·13-s − 3.74i·14-s + 4.00·16-s − 13.2·17-s + (−10.8 − 6.63i)18-s + 5.47·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + (−0.962 − 0.270i)3-s + 0.500·4-s + (0.680 + 0.191i)6-s + 0.377i·7-s − 0.353·8-s + (0.853 + 0.521i)9-s − 1.79i·11-s + (−0.481 − 0.135i)12-s + 1.43i·13-s − 0.267i·14-s + 0.250·16-s − 0.781·17-s + (−0.603 − 0.368i)18-s + 0.288·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8082087998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8082087998\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 + (2.88 + 0.812i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 11 | \( 1 + 19.7iT - 121T^{2} \) |
| 13 | \( 1 - 18.5iT - 169T^{2} \) |
| 17 | \( 1 + 13.2T + 289T^{2} \) |
| 19 | \( 1 - 5.47T + 361T^{2} \) |
| 23 | \( 1 - 34.8T + 529T^{2} \) |
| 29 | \( 1 + 6.83iT - 841T^{2} \) |
| 31 | \( 1 + 31.4T + 961T^{2} \) |
| 37 | \( 1 - 10.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 28.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 19.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 45.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 74.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 32.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 71.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 66.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 101. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 45.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 140.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 100.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 11.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 102. iT - 9.40e3T^{2} \) |
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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
−9.630220612811243293168317550288, −8.954443145048063403718753705070, −8.206614309212645691803364789758, −7.05928358397829141965623472712, −6.47712217472044401177238933683, −5.68248579231974342574176447267, −4.69636860774100342408574698989, −3.32450641019105029734791039566, −1.92709380480213134243566267574, −0.72861648026591030538293254777,
0.57421058308508836370939890645, 1.87457707031054699285923430303, 3.38756935111363421145552180152, 4.68502912596203043960067863731, 5.30714345583585407740582957370, 6.53670003469392887766493092261, 7.17488630404577605843963770292, 7.85773586721262379443679530066, 9.188261654574633579290053175494, 9.721719046499582368378385265709
