L(s) = 1 | − 3·3-s + 5-s + 3·7-s + 6·9-s + 3·11-s − 3·15-s − 7·17-s + 19-s − 9·21-s − 7·23-s + 25-s − 9·27-s − 5·29-s − 4·31-s − 9·33-s + 3·35-s − 3·37-s + 7·41-s − 9·43-s + 6·45-s + 8·47-s + 2·49-s + 21·51-s − 6·53-s + 3·55-s − 3·57-s + 5·59-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 0.447·5-s + 1.13·7-s + 2·9-s + 0.904·11-s − 0.774·15-s − 1.69·17-s + 0.229·19-s − 1.96·21-s − 1.45·23-s + 1/5·25-s − 1.73·27-s − 0.928·29-s − 0.718·31-s − 1.56·33-s + 0.507·35-s − 0.493·37-s + 1.09·41-s − 1.37·43-s + 0.894·45-s + 1.16·47-s + 2/7·49-s + 2.94·51-s − 0.824·53-s + 0.404·55-s − 0.397·57-s + 0.650·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−8.191879225659007797647731332571, −7.22371655291683024694800705470, −6.60809449915252201239251857845, −5.91400312528828131531069367211, −5.31083524266565387680148126615, −4.51008021915387263529049833769, −3.96048949492550969061588178603, −2.08359744541676525405918074810, −1.39912077149714317662152687131, 0,
1.39912077149714317662152687131, 2.08359744541676525405918074810, 3.96048949492550969061588178603, 4.51008021915387263529049833769, 5.31083524266565387680148126615, 5.91400312528828131531069367211, 6.60809449915252201239251857845, 7.22371655291683024694800705470, 8.191879225659007797647731332571