| L(s) = 1 | − 1.41·2-s + (2.53 − 1.59i)3-s + 2.00·4-s + (−3.59 + 2.26i)6-s + 2.64i·7-s − 2.82·8-s + (3.89 − 8.11i)9-s + 7.44i·11-s + (5.07 − 3.19i)12-s + 13.9i·13-s − 3.74i·14-s + 4.00·16-s − 17.8·17-s + (−5.50 + 11.4i)18-s − 33.6·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + (0.846 − 0.532i)3-s + 0.500·4-s + (−0.598 + 0.376i)6-s + 0.377i·7-s − 0.353·8-s + (0.432 − 0.901i)9-s + 0.677i·11-s + (0.423 − 0.266i)12-s + 1.07i·13-s − 0.267i·14-s + 0.250·16-s − 1.05·17-s + (−0.305 + 0.637i)18-s − 1.77·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 - 0.518i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.854 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2748843046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2748843046\) |
| \(L(2)\) |
|
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 | \( 1 + (-2.53 + 1.59i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 11 | \( 1 - 7.44iT - 121T^{2} \) |
| 13 | \( 1 - 13.9iT - 169T^{2} \) |
| 17 | \( 1 + 17.8T + 289T^{2} \) |
| 19 | \( 1 + 33.6T + 361T^{2} \) |
| 23 | \( 1 - 7.94T + 529T^{2} \) |
| 29 | \( 1 + 8.60iT - 841T^{2} \) |
| 31 | \( 1 + 56.7T + 961T^{2} \) |
| 37 | \( 1 - 12.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 75.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 60.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 67.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 49.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 82.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 6.51T + 3.72e3T^{2} \) |
| 67 | \( 1 + 27.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 1.41iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 46.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 49.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 42.4T + 6.88e3T^{2} \) |
| 89 | \( 1 - 63.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 34.3iT - 9.40e3T^{2} \) |
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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
−9.711751754948546443504185154105, −9.097247046272517867446406531058, −8.576146066228244180222438615502, −7.67719748414626880992636075446, −6.77986287383953520108523555868, −6.33637963083695561812430957609, −4.71265949689348194214473295898, −3.68806293462801512630029998009, −2.25396298827134877151434581325, −1.81534281162086740465718928039,
0.085605754702574636708934607317, 1.79526505264948874386734509664, 2.90444315139955380486176426810, 3.83440910091789788271290411844, 4.90909089347550213217017637324, 6.10905187813841240166951123737, 7.10809208810277815665127703648, 7.989372127058237260168003863271, 8.638110148200982430009514241530, 9.196431427393346585045001615896
