L(s) = 1 | + 3-s − 5-s + 9-s + 4·11-s − 13-s − 15-s + 17-s − 4·19-s + 25-s + 27-s + 2·29-s + 4·33-s − 6·37-s − 39-s − 6·41-s − 4·43-s − 45-s − 7·49-s + 51-s − 6·53-s − 4·55-s − 4·57-s + 4·59-s − 6·61-s + 65-s + 12·67-s + 16·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 0.258·15-s + 0.242·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.696·33-s − 0.986·37-s − 0.160·39-s − 0.937·41-s − 0.609·43-s − 0.149·45-s − 49-s + 0.140·51-s − 0.824·53-s − 0.539·55-s − 0.529·57-s + 0.520·59-s − 0.768·61-s + 0.124·65-s + 1.46·67-s + 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.380175475\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.380175475\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−12.91206814488558, −12.47953443008461, −12.20113759890354, −11.57330658208670, −11.25923881718552, −10.62549759369110, −10.17441000449070, −9.610110282718952, −9.287576780016405, −8.655668247958884, −8.282928323643910, −7.984684373258024, −7.143134578707913, −6.881525687470080, −6.407122812462980, −5.866235916250572, −4.975993526485269, −4.745231287877953, −4.038380479268979, −3.539202405626876, −3.241740871166513, −2.384183584936923, −1.832616949236815, −1.271105734402625, −0.4185159062344675,
0.4185159062344675, 1.271105734402625, 1.832616949236815, 2.384183584936923, 3.241740871166513, 3.539202405626876, 4.038380479268979, 4.745231287877953, 4.975993526485269, 5.866235916250572, 6.407122812462980, 6.881525687470080, 7.143134578707913, 7.984684373258024, 8.282928323643910, 8.655668247958884, 9.287576780016405, 9.610110282718952, 10.17441000449070, 10.62549759369110, 11.25923881718552, 11.57330658208670, 12.20113759890354, 12.47953443008461, 12.91206814488558