L(s) = 1 | + (1.20 + 1.59i)2-s − 1.73i·3-s + (−1.10 + 3.84i)4-s − 6.29·5-s + (2.76 − 2.08i)6-s − 2.02i·7-s + (−7.47 + 2.85i)8-s − 2.99·9-s + (−7.56 − 10.0i)10-s − 16.1i·11-s + (6.65 + 1.91i)12-s + 2.97·13-s + (3.23 − 2.43i)14-s + 10.8i·15-s + (−13.5 − 8.49i)16-s − 22.0·17-s + ⋯ |
L(s) = 1 | + (0.601 + 0.798i)2-s − 0.577i·3-s + (−0.276 + 0.961i)4-s − 1.25·5-s + (0.461 − 0.347i)6-s − 0.289i·7-s + (−0.933 + 0.357i)8-s − 0.333·9-s + (−0.756 − 1.00i)10-s − 1.46i·11-s + (0.554 + 0.159i)12-s + 0.228·13-s + (0.231 − 0.174i)14-s + 0.726i·15-s + (−0.847 − 0.531i)16-s − 1.29·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 + 0.961i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.276 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.348727 - 0.463107i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.348727 - 0.463107i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.20 - 1.59i)T \) |
| 3 | \( 1 + 1.73iT \) |
| 23 | \( 1 + 4.79iT \) |
good | 5 | \( 1 + 6.29T + 25T^{2} \) |
| 7 | \( 1 + 2.02iT - 49T^{2} \) |
| 11 | \( 1 + 16.1iT - 121T^{2} \) |
| 13 | \( 1 - 2.97T + 169T^{2} \) |
| 17 | \( 1 + 22.0T + 289T^{2} \) |
| 19 | \( 1 + 25.8iT - 361T^{2} \) |
| 29 | \( 1 + 14.0T + 841T^{2} \) |
| 31 | \( 1 - 56.6iT - 961T^{2} \) |
| 37 | \( 1 + 44.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 26.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 8.26iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 3.15iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 32.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 22.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 81.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 83.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 54.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 6.26T + 5.32e3T^{2} \) |
| 79 | \( 1 + 70.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 10.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 109.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 146.T + 9.40e3T^{2} \) |
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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
−11.49734668239537194323302302849, −10.94480040560677467144743133489, −8.715584332801711850233744391507, −8.489824667142429934970981320404, −7.20783235048476590229014120455, −6.66093613137299452312068129347, −5.29834901264156812813034575429, −4.06751046391296080978188503492, −3.03612803814961410545351677159, −0.23043267259195505425894188968,
2.14874782191698792588569332837, 3.79677443212473698088358601028, 4.31330933518409393224890081859, 5.52633737376235273390556932072, 6.95676425629317980087390102603, 8.236755590502206747260480292102, 9.356846077311393673148598523935, 10.22124323779231331018626199628, 11.18926210468023904956616651811, 11.90400682784115068893819327893