| L(s) = 1 | + 4i·2-s − 8·4-s + 6i·7-s − 32·11-s + 38i·13-s − 24·14-s − 64·16-s − 26i·17-s − 100·19-s − 128i·22-s − 78i·23-s − 152·26-s − 48i·28-s − 50·29-s − 108·31-s − 256i·32-s + ⋯ |
| L(s) = 1 | + 1.41i·2-s − 4-s + 0.323i·7-s − 0.877·11-s + 0.810i·13-s − 0.458·14-s − 16-s − 0.370i·17-s − 1.20·19-s − 1.24i·22-s − 0.707i·23-s − 1.14·26-s − 0.323i·28-s − 0.320·29-s − 0.625·31-s − 1.41i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.372715 - 0.603066i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.372715 - 0.603066i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 4iT - 8T^{2} \) |
| 7 | \( 1 - 6iT - 343T^{2} \) |
| 11 | \( 1 + 32T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 26iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 100T + 6.85e3T^{2} \) |
| 23 | \( 1 + 78iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 50T + 2.43e4T^{2} \) |
| 31 | \( 1 + 108T + 2.97e4T^{2} \) |
| 37 | \( 1 - 266iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 22T + 6.89e4T^{2} \) |
| 43 | \( 1 + 442iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 514iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 2iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 500T + 2.05e5T^{2} \) |
| 61 | \( 1 + 518T + 2.26e5T^{2} \) |
| 67 | \( 1 - 126iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 412T + 3.57e5T^{2} \) |
| 73 | \( 1 - 878iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 600T + 4.93e5T^{2} \) |
| 83 | \( 1 - 282iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 150T + 7.04e5T^{2} \) |
| 97 | \( 1 - 386iT - 9.12e5T^{2} \) |
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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
−12.66464164923169188111478288414, −11.48999159216930931950021529863, −10.42965859123453469199420682920, −9.085699558658319960723887401631, −8.362490277408615677388252496252, −7.32488124350269556007860738421, −6.43856895581418512275817067461, −5.43477343017973845487129109246, −4.37910657868203024511301877637, −2.37348008922852876239774577557,
0.27358447374549899257055219164, 1.91314019147042051215469236949, 3.15903330841775996080496884823, 4.29135622900123821162032689401, 5.69287594106284963827397012071, 7.21177933613361833415980915041, 8.383308826170656975080157514172, 9.555324731457588957912947919611, 10.53499610710502673366448436790, 10.91282225535868211705322114922
