| L(s) = 1 | + 5-s − 4·11-s − 2·13-s − 6·17-s + 4·19-s + 25-s + 2·29-s + 2·37-s − 10·41-s − 4·43-s − 7·49-s + 6·53-s − 4·55-s − 4·59-s + 10·61-s − 2·65-s − 12·67-s − 8·71-s + 10·73-s − 8·79-s + 4·83-s − 6·85-s + 18·89-s + 4·95-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.20·11-s − 0.554·13-s − 1.45·17-s + 0.917·19-s + 1/5·25-s + 0.371·29-s + 0.328·37-s − 1.56·41-s − 0.609·43-s − 49-s + 0.824·53-s − 0.539·55-s − 0.520·59-s + 1.28·61-s − 0.248·65-s − 1.46·67-s − 0.949·71-s + 1.17·73-s − 0.900·79-s + 0.439·83-s − 0.650·85-s + 1.90·89-s + 0.410·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−12.87819765935830, −12.07538906648337, −11.95672562496771, −11.25686823528371, −10.93346843777259, −10.34715596541767, −10.04204400067938, −9.620286936603892, −9.104417439692610, −8.518470504307458, −8.283944328228888, −7.597647918381631, −7.195685444257167, −6.769864692018939, −6.214569605238528, −5.719325586231543, −5.149387315669326, −4.793699814790722, −4.443923334267198, −3.530725885794359, −3.133560044659392, −2.512479052672348, −2.105175135070797, −1.526510788109718, −0.6382363707567199, 0,
0.6382363707567199, 1.526510788109718, 2.105175135070797, 2.512479052672348, 3.133560044659392, 3.530725885794359, 4.443923334267198, 4.793699814790722, 5.149387315669326, 5.719325586231543, 6.214569605238528, 6.769864692018939, 7.195685444257167, 7.597647918381631, 8.283944328228888, 8.518470504307458, 9.104417439692610, 9.620286936603892, 10.04204400067938, 10.34715596541767, 10.93346843777259, 11.25686823528371, 11.95672562496771, 12.07538906648337, 12.87819765935830
