| L(s) = 1 | + (0.540 + 1.30i)2-s + (−1.15 − 1.83i)3-s + (−1.41 + 1.41i)4-s + (2.81 + 2.24i)5-s + (1.77 − 2.50i)6-s + (2.89 + 0.661i)7-s + (−2.61 − 1.08i)8-s + (−0.738 + 1.53i)9-s + (−1.40 + 4.88i)10-s + (−0.943 + 2.69i)11-s + (4.22 + 0.968i)12-s + (−1.89 − 3.93i)13-s + (0.702 + 4.14i)14-s + (0.872 − 7.74i)15-s + (0.00597 − 3.99i)16-s + (−3.41 − 3.41i)17-s + ⋯ |
| L(s) = 1 | + (0.382 + 0.924i)2-s + (−0.666 − 1.06i)3-s + (−0.707 + 0.706i)4-s + (1.25 + 1.00i)5-s + (0.724 − 1.02i)6-s + (1.09 + 0.249i)7-s + (−0.923 − 0.383i)8-s + (−0.246 + 0.511i)9-s + (−0.445 + 1.54i)10-s + (−0.284 + 0.812i)11-s + (1.22 + 0.279i)12-s + (−0.524 − 1.09i)13-s + (0.187 + 1.10i)14-s + (0.225 − 2.00i)15-s + (0.00149 − 0.999i)16-s + (−0.827 − 0.827i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.619 - 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.619 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05969 + 0.513680i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05969 + 0.513680i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.540 - 1.30i)T \) |
| 29 | \( 1 + (1.07 - 5.27i)T \) |
| good | 3 | \( 1 + (1.15 + 1.83i)T + (-1.30 + 2.70i)T^{2} \) |
| 5 | \( 1 + (-2.81 - 2.24i)T + (1.11 + 4.87i)T^{2} \) |
| 7 | \( 1 + (-2.89 - 0.661i)T + (6.30 + 3.03i)T^{2} \) |
| 11 | \( 1 + (0.943 - 2.69i)T + (-8.60 - 6.85i)T^{2} \) |
| 13 | \( 1 + (1.89 + 3.93i)T + (-8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (3.41 + 3.41i)T + 17iT^{2} \) |
| 19 | \( 1 + (0.256 + 0.161i)T + (8.24 + 17.1i)T^{2} \) |
| 23 | \( 1 + (-0.658 + 0.525i)T + (5.11 - 22.4i)T^{2} \) |
| 31 | \( 1 + (0.911 + 8.08i)T + (-30.2 + 6.89i)T^{2} \) |
| 37 | \( 1 + (1.23 - 0.432i)T + (28.9 - 23.0i)T^{2} \) |
| 41 | \( 1 + (-1.66 + 1.66i)T - 41iT^{2} \) |
| 43 | \( 1 + (2.78 + 0.314i)T + (41.9 + 9.56i)T^{2} \) |
| 47 | \( 1 + (-9.56 - 3.34i)T + (36.7 + 29.3i)T^{2} \) |
| 53 | \( 1 + (-3.16 + 3.97i)T + (-11.7 - 51.6i)T^{2} \) |
| 59 | \( 1 - 0.515iT - 59T^{2} \) |
| 61 | \( 1 + (10.6 - 6.71i)T + (26.4 - 54.9i)T^{2} \) |
| 67 | \( 1 + (8.68 + 4.18i)T + (41.7 + 52.3i)T^{2} \) |
| 71 | \( 1 + (2.66 - 1.28i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-12.4 - 1.40i)T + (71.1 + 16.2i)T^{2} \) |
| 79 | \( 1 + (4.20 - 1.47i)T + (61.7 - 49.2i)T^{2} \) |
| 83 | \( 1 + (-10.7 + 2.45i)T + (74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-0.739 + 0.0832i)T + (86.7 - 19.8i)T^{2} \) |
| 97 | \( 1 + (3.88 + 2.44i)T + (42.0 + 87.3i)T^{2} \) |
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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
−13.67342874730830712542606732609, −12.94546028886660028387091432943, −11.92174317699145547055974116888, −10.68171253033599668217247280544, −9.335002937283919035266394558086, −7.68542728772451186505102197354, −7.00424089664403854262224706829, −5.93204240461331684687933880059, −5.04296858559599462096195762390, −2.37441140752796751994621447215,
1.81723095942310868820920913924, 4.31720974380049704633800348602, 5.01739496506010903933358843869, 5.95410359450727594410075723068, 8.616134066291552124016206625658, 9.442353420154659200007417437489, 10.48017528306123817742848656337, 11.13962371253570860966711002551, 12.22310925238603774102687991739, 13.46931201843983878839616812781
