L(s) = 1 | + (−0.149 + 0.988i)3-s + (−0.433 + 0.900i)4-s + (0.294 + 0.955i)7-s + (−0.955 − 0.294i)9-s + (−0.826 − 0.563i)12-s + (−0.997 − 0.0747i)13-s + (−0.623 − 0.781i)16-s + (−1.70 + 0.457i)19-s + (−0.988 + 0.149i)21-s + (−0.294 + 0.955i)25-s + (0.433 − 0.900i)27-s + (−0.988 − 0.149i)28-s + (1.82 − 0.488i)31-s + (0.680 − 0.733i)36-s + (1.43 + 0.500i)37-s + ⋯ |
L(s) = 1 | + (−0.149 + 0.988i)3-s + (−0.433 + 0.900i)4-s + (0.294 + 0.955i)7-s + (−0.955 − 0.294i)9-s + (−0.826 − 0.563i)12-s + (−0.997 − 0.0747i)13-s + (−0.623 − 0.781i)16-s + (−1.70 + 0.457i)19-s + (−0.988 + 0.149i)21-s + (−0.294 + 0.955i)25-s + (0.433 − 0.900i)27-s + (−0.988 − 0.149i)28-s + (1.82 − 0.488i)31-s + (0.680 − 0.733i)36-s + (1.43 + 0.500i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6201042213\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6201042213\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.149 - 0.988i)T \) |
| 7 | \( 1 + (-0.294 - 0.955i)T \) |
| 13 | \( 1 + (0.997 + 0.0747i)T \) |
good | 2 | \( 1 + (0.433 - 0.900i)T^{2} \) |
| 5 | \( 1 + (0.294 - 0.955i)T^{2} \) |
| 11 | \( 1 + (-0.563 - 0.826i)T^{2} \) |
| 17 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 19 | \( 1 + (1.70 - 0.457i)T + (0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 29 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 31 | \( 1 + (-1.82 + 0.488i)T + (0.866 - 0.5i)T^{2} \) |
| 37 | \( 1 + (-1.43 - 0.500i)T + (0.781 + 0.623i)T^{2} \) |
| 41 | \( 1 + (0.680 + 0.733i)T^{2} \) |
| 43 | \( 1 + (0.548 + 0.215i)T + (0.733 + 0.680i)T^{2} \) |
| 47 | \( 1 + (0.563 + 0.826i)T^{2} \) |
| 53 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 59 | \( 1 + (-0.974 - 0.222i)T^{2} \) |
| 61 | \( 1 + (1.64 + 0.123i)T + (0.988 + 0.149i)T^{2} \) |
| 67 | \( 1 + (0.216 + 0.0579i)T + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + (-0.149 - 0.988i)T^{2} \) |
| 73 | \( 1 + (1.73 - 0.918i)T + (0.563 - 0.826i)T^{2} \) |
| 79 | \( 1 + (-0.433 - 0.751i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.433 + 0.900i)T^{2} \) |
| 89 | \( 1 + (-0.433 - 0.900i)T^{2} \) |
| 97 | \( 1 + (0.241 - 0.902i)T + (-0.866 - 0.5i)T^{2} \) |
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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
−9.670146061857479031401649253131, −9.062730546000729408612122890810, −8.324055652808701668952446694338, −7.81834497253582632878291628200, −6.52805598357093231853489021540, −5.68393098430927786156137137301, −4.70275304530923340427064502626, −4.28461334063412520248967784509, −3.10519422745805051960735440313, −2.35784046449394394281270542327,
0.45047638482190693969591226831, 1.69336640879571032071954416015, 2.69349855733416730872155226352, 4.40891932209757719852341080981, 4.74608161450532793598690898208, 6.04882160497446264901633820305, 6.49524142188700750255251870208, 7.36386589262847452049484690244, 8.156116180967920300099512592304, 8.857177210436208552281725843020