L(s) = 1 | + 5-s − 5·11-s − 2·13-s + 6·17-s − 2·19-s − 6·23-s − 4·25-s + 3·29-s + 5·31-s + 2·37-s + 8·41-s + 4·43-s + 4·47-s − 9·53-s − 5·55-s + 3·59-s + 12·61-s − 2·65-s − 2·67-s − 8·71-s − 14·73-s + 79-s − 17·83-s + 6·85-s + 18·89-s − 2·95-s + 3·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.50·11-s − 0.554·13-s + 1.45·17-s − 0.458·19-s − 1.25·23-s − 4/5·25-s + 0.557·29-s + 0.898·31-s + 0.328·37-s + 1.24·41-s + 0.609·43-s + 0.583·47-s − 1.23·53-s − 0.674·55-s + 0.390·59-s + 1.53·61-s − 0.248·65-s − 0.244·67-s − 0.949·71-s − 1.63·73-s + 0.112·79-s − 1.86·83-s + 0.650·85-s + 1.90·89-s − 0.205·95-s + 0.304·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−15.62586699175365, −14.88298689661112, −14.23731824817517, −14.09636774489458, −13.23063957483642, −12.91833101118430, −12.28290913221122, −11.84281610659255, −11.18596632926422, −10.32120787860158, −10.16979018989713, −9.747913861494221, −8.947947392632356, −8.217493358815916, −7.722677955323440, −7.461933785446865, −6.450636861802114, −5.791718041154936, −5.560570495789326, −4.700768781083976, −4.166825877087931, −3.221453329334809, −2.576298669841226, −2.061352715003202, −1.001536256859518, 0,
1.001536256859518, 2.061352715003202, 2.576298669841226, 3.221453329334809, 4.166825877087931, 4.700768781083976, 5.560570495789326, 5.791718041154936, 6.450636861802114, 7.461933785446865, 7.722677955323440, 8.217493358815916, 8.947947392632356, 9.747913861494221, 10.16979018989713, 10.32120787860158, 11.18596632926422, 11.84281610659255, 12.28290913221122, 12.91833101118430, 13.23063957483642, 14.09636774489458, 14.23731824817517, 14.88298689661112, 15.62586699175365