| L(s) = 1 | + (1.90 − 0.598i)2-s + (3.28 − 2.28i)4-s + 2.62·5-s + 0.432·7-s + (4.90 − 6.32i)8-s + (5 − 1.56i)10-s + 5.01·11-s + 1.56i·13-s + (0.824 − 0.258i)14-s + (5.56 − 15i)16-s − 19.8i·17-s + 24.1i·19-s + (8.60 − 5.98i)20-s + (9.56 − 3i)22-s + 3.07i·23-s + ⋯ |
| L(s) = 1 | + (0.954 − 0.299i)2-s + (0.820 − 0.570i)4-s + 0.524·5-s + 0.0617·7-s + (0.612 − 0.790i)8-s + (0.5 − 0.156i)10-s + 0.455·11-s + 0.120i·13-s + (0.0589 − 0.0184i)14-s + (0.347 − 0.937i)16-s − 1.16i·17-s + 1.27i·19-s + (0.430 − 0.299i)20-s + (0.434 − 0.136i)22-s + 0.133i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.81200 - 0.962028i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.81200 - 0.962028i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.90 + 0.598i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.62T + 25T^{2} \) |
| 7 | \( 1 - 0.432T + 49T^{2} \) |
| 11 | \( 1 - 5.01T + 121T^{2} \) |
| 13 | \( 1 - 1.56iT - 169T^{2} \) |
| 17 | \( 1 + 19.8iT - 289T^{2} \) |
| 19 | \( 1 - 24.1iT - 361T^{2} \) |
| 23 | \( 1 - 3.07iT - 529T^{2} \) |
| 29 | \( 1 - 34.4T + 841T^{2} \) |
| 31 | \( 1 + 43.4T + 961T^{2} \) |
| 37 | \( 1 - 52.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 56.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 27.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 83.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 41.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 74.2T + 3.48e3T^{2} \) |
| 61 | \( 1 - 28.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 33.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 104. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 53.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 51.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + 76.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + 131. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 68.9T + 9.40e3T^{2} \) |
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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
−12.06924634126933970734396522256, −11.25594588680100739624012984970, −10.15148994838746466711609248554, −9.351309102997880296828598897722, −7.75598745434720319650357635618, −6.56788533262402122941615897516, −5.63788357894166826605974697163, −4.48148501121599917487003800440, −3.13421307425443634619829194095, −1.62515203115120789496786062332,
2.01738807565393690129839082918, 3.55660315928120250603567590713, 4.81528856590780194565594642068, 5.95246014544108057921199406088, 6.81341601186862898972070676813, 8.019686732049431757860608073844, 9.174552496715545708932020482092, 10.50547808939254851876257006094, 11.36392978241858288400772496917, 12.43052230215211933631685528180
