L(s) = 1 | + i·2-s − 4-s + (2.09 − 3.62i)5-s + (2.62 − 0.358i)7-s − i·8-s + (3.62 + 2.09i)10-s + (2.59 − 1.5i)11-s + (−2.12 + 1.22i)13-s + (0.358 + 2.62i)14-s + 16-s + (−0.507 + 0.878i)17-s + (−0.878 + 0.507i)19-s + (−2.09 + 3.62i)20-s + (1.5 + 2.59i)22-s + (3.67 + 2.12i)23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (0.935 − 1.61i)5-s + (0.990 − 0.135i)7-s − 0.353i·8-s + (1.14 + 0.661i)10-s + (0.783 − 0.452i)11-s + (−0.588 + 0.339i)13-s + (0.0958 + 0.700i)14-s + 0.250·16-s + (−0.123 + 0.213i)17-s + (−0.201 + 0.116i)19-s + (−0.467 + 0.809i)20-s + (0.319 + 0.553i)22-s + (0.766 + 0.442i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.023517576\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.023517576\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.62 + 0.358i)T \) |
good | 5 | \( 1 + (-2.09 + 3.62i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.12 - 1.22i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.507 - 0.878i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.878 - 0.507i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.67 - 2.12i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.07 - 0.621i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 5.61iT - 31T^{2} \) |
| 37 | \( 1 + (4.12 + 7.13i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.01 + 1.75i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.12 - 7.13i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 1.01T + 47T^{2} \) |
| 53 | \( 1 + (-1.07 - 0.621i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 - 5.91iT - 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 + 10.2iT - 71T^{2} \) |
| 73 | \( 1 + (-7.24 - 4.18i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 + (-1.58 + 2.74i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.19 - 9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.25 - 1.88i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.381993264673161030249356595844, −8.914530585412244381533456969047, −8.271889445561445792398737531090, −7.34891324781678848848010287674, −6.23022355854124706435367024780, −5.44313591373444305418836203375, −4.79645508841850714777518476376, −4.01293905825042585200801203311, −2.00197412349096806972023762743, −0.980411682521153323900069134756,
1.61677208438451044090099080924, 2.48116113090829759941842861093, 3.36212926535160922796774501589, 4.66633461835822587684707927957, 5.51972678500407866692770604530, 6.69060614815535879890166818569, 7.17958797887611984017895073560, 8.399895300320080320385674861909, 9.266405376959575739430538673763, 10.21053812304436736248063465004