| L(s) = 1 | + (1.30 + 0.545i)2-s + (0.964 + 1.43i)3-s + (1.40 + 1.42i)4-s + (−2.14 − 0.638i)5-s + (0.473 + 2.40i)6-s + 4.06·7-s + (1.05 + 2.62i)8-s + (−1.13 + 2.77i)9-s + (−2.44 − 2.00i)10-s + (−2.61 − 1.89i)11-s + (−0.691 + 3.39i)12-s + (−3.55 − 4.88i)13-s + (5.30 + 2.21i)14-s + (−1.14 − 3.69i)15-s + (−0.0500 + 3.99i)16-s + (−0.528 + 1.62i)17-s + ⋯ |
| L(s) = 1 | + (0.922 + 0.385i)2-s + (0.556 + 0.830i)3-s + (0.702 + 0.711i)4-s + (−0.958 − 0.285i)5-s + (0.193 + 0.981i)6-s + 1.53·7-s + (0.373 + 0.927i)8-s + (−0.379 + 0.925i)9-s + (−0.774 − 0.632i)10-s + (−0.787 − 0.571i)11-s + (−0.199 + 0.979i)12-s + (−0.985 − 1.35i)13-s + (1.41 + 0.592i)14-s + (−0.296 − 0.954i)15-s + (−0.0125 + 0.999i)16-s + (−0.128 + 0.394i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.273 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.273 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.90749 + 1.44034i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.90749 + 1.44034i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.30 - 0.545i)T \) |
| 3 | \( 1 + (-0.964 - 1.43i)T \) |
| 5 | \( 1 + (2.14 + 0.638i)T \) |
| good | 7 | \( 1 - 4.06T + 7T^{2} \) |
| 11 | \( 1 + (2.61 + 1.89i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (3.55 + 4.88i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.528 - 1.62i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.49 - 0.809i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-2.71 + 3.74i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.01 + 0.655i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.77 + 0.576i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (1.36 + 1.88i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (4.31 + 5.94i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 5.20T + 43T^{2} \) |
| 47 | \( 1 + (-3.27 + 1.06i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-2.20 - 6.78i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-9.81 + 7.13i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (4.99 + 3.62i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-0.983 + 3.02i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-1.67 - 5.15i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (1.58 - 2.17i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-2.42 + 0.787i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (5.77 + 1.87i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (4.38 - 6.04i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (14.9 - 4.85i)T + (78.4 - 57.0i)T^{2} \) |
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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
−11.97165099441857040443631998296, −11.02652533673722374196214452897, −10.45514114674059899161996254776, −8.590839381770588886565111050917, −8.062968841193650190995836266178, −7.42944439944502719954851707168, −5.28651687557262684666963314185, −4.98239798835016589562952120912, −3.79091110509395602705776337496, −2.62531105005601328189206664213,
1.70228967428304444504632662526, 2.87764318941764982643743106452, 4.35820795752530193157436574678, 5.17470918715624651064251804868, 7.03259634072468909589569605135, 7.33908202302510214848191781999, 8.440562033974021882144833012735, 9.795791887348435387558292371116, 11.19077246155630386289212476571, 11.69911761109038734225468017495
