| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 11-s + 2·13-s + 16-s + 5·17-s − 6·19-s − 20-s − 22-s + 7·23-s − 4·25-s + 2·26-s − 8·29-s + 10·31-s + 32-s + 5·34-s − 8·37-s − 6·38-s − 40-s + 7·41-s + 4·43-s − 44-s + 7·46-s − 47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 0.301·11-s + 0.554·13-s + 1/4·16-s + 1.21·17-s − 1.37·19-s − 0.223·20-s − 0.213·22-s + 1.45·23-s − 4/5·25-s + 0.392·26-s − 1.48·29-s + 1.79·31-s + 0.176·32-s + 0.857·34-s − 1.31·37-s − 0.973·38-s − 0.158·40-s + 1.09·41-s + 0.609·43-s − 0.150·44-s + 1.03·46-s − 0.145·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.080189968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.080189968\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−7.47068797961057907569044063720, −7.08214782027466981971467646227, −6.11491795017896052376336986879, −5.68753040471390287599978752467, −4.86853162491890491224024500651, −4.14755395324560442677419262117, −3.53440920110557896949922156104, −2.78566396672986772511030956709, −1.85732447935128858961460953936, −0.75251221911006198014233530591,
0.75251221911006198014233530591, 1.85732447935128858961460953936, 2.78566396672986772511030956709, 3.53440920110557896949922156104, 4.14755395324560442677419262117, 4.86853162491890491224024500651, 5.68753040471390287599978752467, 6.11491795017896052376336986879, 7.08214782027466981971467646227, 7.47068797961057907569044063720
