| L(s) = 1 | + (−0.841 − 0.540i)2-s + (−0.459 − 3.19i)3-s + (0.415 + 0.909i)4-s + (−1.51 − 0.444i)5-s + (−1.33 + 2.93i)6-s + (−0.654 + 0.755i)7-s + (0.142 − 0.989i)8-s + (−7.10 + 2.08i)9-s + (1.03 + 1.19i)10-s + (−3.31 + 2.13i)11-s + (2.71 − 1.74i)12-s + (1.69 + 1.96i)13-s + (0.959 − 0.281i)14-s + (−0.725 + 5.04i)15-s + (−0.654 + 0.755i)16-s + (2.89 − 6.34i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 − 0.382i)2-s + (−0.265 − 1.84i)3-s + (0.207 + 0.454i)4-s + (−0.677 − 0.199i)5-s + (−0.547 + 1.19i)6-s + (−0.247 + 0.285i)7-s + (0.0503 − 0.349i)8-s + (−2.36 + 0.695i)9-s + (0.327 + 0.377i)10-s + (−1.00 + 0.642i)11-s + (0.783 − 0.503i)12-s + (0.471 + 0.544i)13-s + (0.256 − 0.0752i)14-s + (−0.187 + 1.30i)15-s + (−0.163 + 0.188i)16-s + (0.702 − 1.53i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.124285 + 0.169998i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.124285 + 0.169998i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.841 + 0.540i)T \) |
| 7 | \( 1 + (0.654 - 0.755i)T \) |
| 23 | \( 1 + (3.68 - 3.07i)T \) |
| good | 3 | \( 1 + (0.459 + 3.19i)T + (-2.87 + 0.845i)T^{2} \) |
| 5 | \( 1 + (1.51 + 0.444i)T + (4.20 + 2.70i)T^{2} \) |
| 11 | \( 1 + (3.31 - 2.13i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.69 - 1.96i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.89 + 6.34i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (0.897 + 1.96i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-2.53 + 5.55i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (1.46 - 10.1i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-1.02 + 0.301i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (9.85 + 2.89i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (0.247 + 1.71i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 6.77T + 47T^{2} \) |
| 53 | \( 1 + (2.61 - 3.01i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (0.713 + 0.823i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.25 + 8.72i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (4.61 + 2.96i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (3.57 + 2.29i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.57 - 3.45i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (5.51 + 6.36i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-6.31 + 1.85i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (1.88 + 13.1i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-0.931 - 0.273i)T + (81.6 + 52.4i)T^{2} \) |
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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
−11.41477061137389269354945480435, −10.08937076945148331570860168287, −8.818230111983197246488347963073, −7.923561114496759871293001445539, −7.34882962410105986431577830521, −6.44067084804634699379322587389, −5.09418439502694511603644100463, −3.04119851372408563461313535090, −1.82787598145599978124232362482, −0.18046268199487933902645066696,
3.25367745026461105393073844039, 4.09551532804669460197860299703, 5.44073483740310943916903028234, 6.18076995822366394135768855022, 8.024318116523122140180974696881, 8.399384519669627575869088798510, 9.733256822921316082931910700437, 10.41778853836670527341582303700, 10.84277664715194257136011694903, 11.81101302766857464686556504803
