| L(s) = 1 | + (−0.923 − 1.07i)2-s + (1.20 + 1.24i)3-s + (−0.293 + 1.97i)4-s + (2.16 − 0.562i)5-s + (0.216 − 2.43i)6-s + 3.72i·7-s + (2.38 − 1.51i)8-s + (−0.0905 + 2.99i)9-s + (−2.60 − 1.79i)10-s + (−2.85 + 2.07i)11-s + (−2.81 + 2.02i)12-s + (−2.64 − 1.92i)13-s + (3.99 − 3.44i)14-s + (3.30 + 2.01i)15-s + (−3.82 − 1.16i)16-s + (−1.89 + 0.614i)17-s + ⋯ |
| L(s) = 1 | + (−0.653 − 0.757i)2-s + (0.696 + 0.717i)3-s + (−0.146 + 0.989i)4-s + (0.967 − 0.251i)5-s + (0.0885 − 0.996i)6-s + 1.40i·7-s + (0.844 − 0.535i)8-s + (−0.0301 + 0.999i)9-s + (−0.822 − 0.568i)10-s + (−0.860 + 0.625i)11-s + (−0.812 + 0.583i)12-s + (−0.733 − 0.532i)13-s + (1.06 − 0.920i)14-s + (0.854 + 0.519i)15-s + (−0.957 − 0.290i)16-s + (−0.458 + 0.148i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.19088 + 0.387583i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19088 + 0.387583i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.923 + 1.07i)T \) |
| 3 | \( 1 + (-1.20 - 1.24i)T \) |
| 5 | \( 1 + (-2.16 + 0.562i)T \) |
| good | 7 | \( 1 - 3.72iT - 7T^{2} \) |
| 11 | \( 1 + (2.85 - 2.07i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (2.64 + 1.92i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.89 - 0.614i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-3.02 + 0.982i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.00 + 0.728i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.72 - 0.885i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-9.25 + 3.00i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (2.96 + 2.15i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-7.18 + 9.89i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 5.08iT - 43T^{2} \) |
| 47 | \( 1 + (0.623 - 1.91i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-4.01 - 1.30i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-2.52 - 1.83i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-7.65 + 5.56i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (13.7 - 4.47i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.64 - 5.06i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (5.62 - 4.08i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (9.60 + 3.11i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (4.65 + 14.3i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-1.86 - 2.57i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.754 + 2.32i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.81128016123044269270957773925, −10.45161472981357451207551881287, −9.981538166269195225114431842009, −9.081799555661824258339299785091, −8.542269234254162862925222296577, −7.38989679355358271263188255800, −5.55664225574764154144019055313, −4.63858536506029948691602747836, −2.76273585483310771021322449530, −2.27614608553355393510137126113,
1.15551655276650615036594429986, 2.76078162100792730680600102641, 4.70637192141796172809351170651, 6.13566647143631661900042815470, 6.96482361292718984090611901867, 7.67748471318321512347369743604, 8.651673421254979289396755896342, 9.774871976658715023084863161825, 10.26019132361180753745369622887, 11.43264066578770964461924792981
