L(s) = 1 | + 3·7-s − 5·11-s + 13-s + 5·17-s + 4·19-s − 2·23-s + 9·29-s + 3·31-s − 10·37-s + 12·41-s + 2·43-s − 9·47-s + 2·49-s − 9·53-s − 3·59-s − 7·61-s + 9·67-s − 10·73-s − 15·77-s − 10·79-s − 83-s + 4·89-s + 3·91-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | + 1.13·7-s − 1.50·11-s + 0.277·13-s + 1.21·17-s + 0.917·19-s − 0.417·23-s + 1.67·29-s + 0.538·31-s − 1.64·37-s + 1.87·41-s + 0.304·43-s − 1.31·47-s + 2/7·49-s − 1.23·53-s − 0.390·59-s − 0.896·61-s + 1.09·67-s − 1.17·73-s − 1.70·77-s − 1.12·79-s − 0.109·83-s + 0.423·89-s + 0.314·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.13541958612587, −13.71892528351200, −13.11267685494159, −12.41265135481380, −12.26595117591740, −11.53644074848006, −11.15802218732291, −10.58478505476801, −10.14347942011761, −9.794359924282775, −9.013285478006598, −8.400286577840413, −7.970986482909938, −7.716028744846276, −7.174050415749577, −6.344598175415043, −5.807319254211908, −5.226580502781321, −4.874444935503555, −4.369403332885642, −3.439038229374348, −2.966111042087763, −2.383491187512321, −1.499006771160341, −1.047777486491187, 0,
1.047777486491187, 1.499006771160341, 2.383491187512321, 2.966111042087763, 3.439038229374348, 4.369403332885642, 4.874444935503555, 5.226580502781321, 5.807319254211908, 6.344598175415043, 7.174050415749577, 7.716028744846276, 7.970986482909938, 8.400286577840413, 9.013285478006598, 9.794359924282775, 10.14347942011761, 10.58478505476801, 11.15802218732291, 11.53644074848006, 12.26595117591740, 12.41265135481380, 13.11267685494159, 13.71892528351200, 14.13541958612587