L(s) = 1 | + 7-s + 2·11-s + 13-s + 3·17-s + 23-s + 5·29-s − 7·31-s + 2·37-s − 7·41-s − 11·43-s − 8·47-s + 49-s − 53-s − 5·59-s − 3·61-s − 12·67-s + 12·71-s + 6·73-s + 2·77-s − 10·79-s + 11·83-s + 10·89-s + 91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | + 0.377·7-s + 0.603·11-s + 0.277·13-s + 0.727·17-s + 0.208·23-s + 0.928·29-s − 1.25·31-s + 0.328·37-s − 1.09·41-s − 1.67·43-s − 1.16·47-s + 1/7·49-s − 0.137·53-s − 0.650·59-s − 0.384·61-s − 1.46·67-s + 1.42·71-s + 0.702·73-s + 0.227·77-s − 1.12·79-s + 1.20·83-s + 1.05·89-s + 0.104·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−15.48416364685885, −15.10216758566478, −14.47538182273888, −14.18054583300422, −13.41426360094808, −13.07647326983004, −12.21650353302635, −11.93625436344915, −11.33422648181810, −10.75459337588866, −10.19881654309829, −9.596748712629114, −9.037935108357556, −8.417889706148972, −7.923249998197572, −7.299165408272665, −6.532203670535559, −6.222128411321942, −5.146639241461693, −5.016039671221181, −3.985228629775805, −3.479701765640175, −2.758425435362372, −1.713561925040990, −1.238067897615418, 0,
1.238067897615418, 1.713561925040990, 2.758425435362372, 3.479701765640175, 3.985228629775805, 5.016039671221181, 5.146639241461693, 6.222128411321942, 6.532203670535559, 7.299165408272665, 7.923249998197572, 8.417889706148972, 9.037935108357556, 9.596748712629114, 10.19881654309829, 10.75459337588866, 11.33422648181810, 11.93625436344915, 12.21650353302635, 13.07647326983004, 13.41426360094808, 14.18054583300422, 14.47538182273888, 15.10216758566478, 15.48416364685885