| L(s) = 1 | + (−0.192 + 0.0980i)3-s + (1.96 + 1.07i)5-s + (−3.64 + 3.64i)7-s + (−1.73 + 2.38i)9-s + (−1.40 − 1.93i)11-s + (−3.87 + 0.613i)13-s + (−0.482 − 0.0148i)15-s + (1.70 − 3.34i)17-s + (1.33 + 4.11i)19-s + (0.343 − 1.05i)21-s + (−4.03 − 0.639i)23-s + (2.68 + 4.21i)25-s + (0.201 − 1.26i)27-s + (6.80 + 2.20i)29-s + (4.89 − 1.59i)31-s + ⋯ |
| L(s) = 1 | + (−0.111 + 0.0565i)3-s + (0.876 + 0.481i)5-s + (−1.37 + 1.37i)7-s + (−0.578 + 0.796i)9-s + (−0.424 − 0.584i)11-s + (−1.07 + 0.170i)13-s + (−0.124 − 0.00382i)15-s + (0.413 − 0.812i)17-s + (0.306 + 0.943i)19-s + (0.0749 − 0.230i)21-s + (−0.842 − 0.133i)23-s + (0.537 + 0.843i)25-s + (0.0386 − 0.244i)27-s + (1.26 + 0.410i)29-s + (0.879 − 0.285i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.464 - 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.464 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.490957 + 0.811967i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.490957 + 0.811967i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.96 - 1.07i)T \) |
| good | 3 | \( 1 + (0.192 - 0.0980i)T + (1.76 - 2.42i)T^{2} \) |
| 7 | \( 1 + (3.64 - 3.64i)T - 7iT^{2} \) |
| 11 | \( 1 + (1.40 + 1.93i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (3.87 - 0.613i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-1.70 + 3.34i)T + (-9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (-1.33 - 4.11i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (4.03 + 0.639i)T + (21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (-6.80 - 2.20i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-4.89 + 1.59i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.32 - 8.37i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (3.51 + 2.55i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (0.236 + 0.236i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.73 - 7.32i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-1.46 - 2.87i)T + (-31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-8.27 - 6.00i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (5.77 - 4.19i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-3.40 - 1.73i)T + (39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (-10.6 - 3.45i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.176 + 1.11i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (-3.53 + 10.8i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-6.66 + 13.0i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (0.149 + 0.205i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (6.59 - 3.36i)T + (57.0 - 78.4i)T^{2} \) |
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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
−11.78788880211207894051051277366, −10.33349270770347661858812879584, −9.905857023192647990112808535810, −8.990920953364473774978607268077, −7.944263409265124955761695750443, −6.59153745980643164111309905551, −5.83467445561094880016313537029, −5.12551714404834645827431564419, −2.98370453659624511517897089010, −2.48757310148778413200863072359,
0.59437132838419273282049365240, 2.63418063855943852100036151560, 3.92634671519844497422790413152, 5.18733489179643049442935908911, 6.35795168897627826509038388290, 6.97254699264509799534039329897, 8.214170656326518030364656134509, 9.619734709280989017894907467327, 9.817894882401000682049241237130, 10.69675907668725827359205300434
