L(s) = 1 | − 0.642·3-s − 2.00·5-s − 3.04·7-s − 2.58·9-s − 5.21·11-s + 2.25·13-s + 1.28·15-s + 6.39·17-s + 0.375·19-s + 1.95·21-s + 3.84·23-s − 0.981·25-s + 3.59·27-s + 9.60·31-s + 3.35·33-s + 6.10·35-s − 2.61·37-s − 1.44·39-s − 9.89·41-s + 9.78·43-s + 5.18·45-s + 5.18·47-s + 2.26·49-s − 4.10·51-s + 5.80·53-s + 10.4·55-s − 0.241·57-s + ⋯ |
L(s) = 1 | − 0.371·3-s − 0.896·5-s − 1.15·7-s − 0.862·9-s − 1.57·11-s + 0.624·13-s + 0.332·15-s + 1.55·17-s + 0.0860·19-s + 0.426·21-s + 0.801·23-s − 0.196·25-s + 0.691·27-s + 1.72·31-s + 0.583·33-s + 1.03·35-s − 0.430·37-s − 0.231·39-s − 1.54·41-s + 1.49·43-s + 0.773·45-s + 0.755·47-s + 0.323·49-s − 0.575·51-s + 0.797·53-s + 1.41·55-s − 0.0319·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 \) |
good | 3 | \( 1 + 0.642T + 3T^{2} \) |
| 5 | \( 1 + 2.00T + 5T^{2} \) |
| 7 | \( 1 + 3.04T + 7T^{2} \) |
| 11 | \( 1 + 5.21T + 11T^{2} \) |
| 13 | \( 1 - 2.25T + 13T^{2} \) |
| 17 | \( 1 - 6.39T + 17T^{2} \) |
| 19 | \( 1 - 0.375T + 19T^{2} \) |
| 23 | \( 1 - 3.84T + 23T^{2} \) |
| 31 | \( 1 - 9.60T + 31T^{2} \) |
| 37 | \( 1 + 2.61T + 37T^{2} \) |
| 41 | \( 1 + 9.89T + 41T^{2} \) |
| 43 | \( 1 - 9.78T + 43T^{2} \) |
| 47 | \( 1 - 5.18T + 47T^{2} \) |
| 53 | \( 1 - 5.80T + 53T^{2} \) |
| 59 | \( 1 - 1.95T + 59T^{2} \) |
| 61 | \( 1 - 3.10T + 61T^{2} \) |
| 67 | \( 1 + 2.51T + 67T^{2} \) |
| 71 | \( 1 - 3.72T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 5.15T + 79T^{2} \) |
| 83 | \( 1 - 1.57T + 83T^{2} \) |
| 89 | \( 1 + 17.9T + 89T^{2} \) |
| 97 | \( 1 + 4.11T + 97T^{2} \) |
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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
−7.63710621442515590985379732605, −7.00781539839974707957938736767, −6.07598638373295035716051343967, −5.60059102566770543918531356673, −4.89865819960910261831071265750, −3.83265832172447508666397067215, −3.11216743919226809889998073909, −2.67520800664267799665931226151, −0.913695907106670338989976202569, 0,
0.913695907106670338989976202569, 2.67520800664267799665931226151, 3.11216743919226809889998073909, 3.83265832172447508666397067215, 4.89865819960910261831071265750, 5.60059102566770543918531356673, 6.07598638373295035716051343967, 7.00781539839974707957938736767, 7.63710621442515590985379732605