| L(s) = 1 | + (0.998 − 0.0627i)2-s + (−0.904 + 0.425i)3-s + (0.992 − 0.125i)4-s + (1.89 − 1.19i)5-s + (−0.876 + 0.481i)6-s + (2.52 + 3.47i)7-s + (0.982 − 0.187i)8-s + (0.637 − 0.770i)9-s + (1.81 − 1.30i)10-s + (0.185 + 2.94i)11-s + (−0.844 + 0.535i)12-s + (−2.06 − 1.71i)13-s + (2.73 + 3.30i)14-s + (−1.20 + 1.88i)15-s + (0.968 − 0.248i)16-s + (0.0331 − 0.262i)17-s + ⋯ |
| L(s) = 1 | + (0.705 − 0.0443i)2-s + (−0.522 + 0.245i)3-s + (0.496 − 0.0626i)4-s + (0.846 − 0.532i)5-s + (−0.357 + 0.196i)6-s + (0.953 + 1.31i)7-s + (0.347 − 0.0662i)8-s + (0.212 − 0.256i)9-s + (0.573 − 0.413i)10-s + (0.0558 + 0.888i)11-s + (−0.243 + 0.154i)12-s + (−0.574 − 0.474i)13-s + (0.730 + 0.883i)14-s + (−0.310 + 0.486i)15-s + (0.242 − 0.0621i)16-s + (0.00804 − 0.0636i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.39488 + 0.618094i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.39488 + 0.618094i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.998 + 0.0627i)T \) |
| 3 | \( 1 + (0.904 - 0.425i)T \) |
| 5 | \( 1 + (-1.89 + 1.19i)T \) |
| good | 7 | \( 1 + (-2.52 - 3.47i)T + (-2.16 + 6.65i)T^{2} \) |
| 11 | \( 1 + (-0.185 - 2.94i)T + (-10.9 + 1.37i)T^{2} \) |
| 13 | \( 1 + (2.06 + 1.71i)T + (2.43 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-0.0331 + 0.262i)T + (-16.4 - 4.22i)T^{2} \) |
| 19 | \( 1 + (1.89 - 4.01i)T + (-12.1 - 14.6i)T^{2} \) |
| 23 | \( 1 + (0.891 + 2.25i)T + (-16.7 + 15.7i)T^{2} \) |
| 29 | \( 1 + (-4.44 + 4.17i)T + (1.82 - 28.9i)T^{2} \) |
| 31 | \( 1 + (-1.23 - 0.155i)T + (30.0 + 7.70i)T^{2} \) |
| 37 | \( 1 + (-1.68 - 6.55i)T + (-32.4 + 17.8i)T^{2} \) |
| 41 | \( 1 + (-3.23 - 1.28i)T + (29.8 + 28.0i)T^{2} \) |
| 43 | \( 1 + (1.43 + 0.465i)T + (34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (5.74 + 1.09i)T + (43.6 + 17.3i)T^{2} \) |
| 53 | \( 1 + (-4.89 + 8.89i)T + (-28.3 - 44.7i)T^{2} \) |
| 59 | \( 1 + (-4.49 - 7.07i)T + (-25.1 + 53.3i)T^{2} \) |
| 61 | \( 1 + (-0.0151 + 0.00600i)T + (44.4 - 41.7i)T^{2} \) |
| 67 | \( 1 + (-4.88 + 5.20i)T + (-4.20 - 66.8i)T^{2} \) |
| 71 | \( 1 + (-0.522 + 2.73i)T + (-66.0 - 26.1i)T^{2} \) |
| 73 | \( 1 + (11.8 + 7.54i)T + (31.0 + 66.0i)T^{2} \) |
| 79 | \( 1 + (6.18 + 13.1i)T + (-50.3 + 60.8i)T^{2} \) |
| 83 | \( 1 + (2.19 + 1.03i)T + (52.9 + 63.9i)T^{2} \) |
| 89 | \( 1 + (4.00 - 6.31i)T + (-37.8 - 80.5i)T^{2} \) |
| 97 | \( 1 + (7.32 + 7.79i)T + (-6.09 + 96.8i)T^{2} \) |
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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
−10.30870427944967513609112656549, −9.839798829286553299429018288640, −8.700762457691403717201797888131, −7.906168249227583171305126333999, −6.51238166028199761075633533956, −5.76256624170860252858886197483, −5.01649505857772135294694672430, −4.44095123595499540279572225017, −2.61227570346718349614129765529, −1.70103123144893924389940431796,
1.25673593008518745752578556260, 2.58065852689916181424852747258, 3.98990347040473422990842185396, 4.91556091703515843536256905432, 5.78646743697221973141733006769, 6.81120505978160630743717087243, 7.26351997243997114331476102423, 8.405167493588826433655666121823, 9.688722630942379188609743661292, 10.68621225521620741642106795909
