| L(s) = 1 | − 5-s − 7-s + 8·19-s − 3·23-s − 4·25-s + 8·29-s − 31-s + 35-s + 3·37-s + 8·41-s + 8·43-s + 4·47-s + 49-s − 6·53-s − 9·59-s + 8·61-s − 3·67-s + 7·71-s − 8·73-s + 8·83-s − 7·89-s − 8·95-s − 7·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 1.83·19-s − 0.625·23-s − 4/5·25-s + 1.48·29-s − 0.179·31-s + 0.169·35-s + 0.493·37-s + 1.24·41-s + 1.21·43-s + 0.583·47-s + 1/7·49-s − 0.824·53-s − 1.17·59-s + 1.02·61-s − 0.366·67-s + 0.830·71-s − 0.936·73-s + 0.878·83-s − 0.741·89-s − 0.820·95-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.76203882940716, −13.45441079664826, −12.72401735835964, −12.15574681653228, −12.07249880803757, −11.38246098475865, −10.94888792769853, −10.40699647735543, −9.780977750093730, −9.461969348782195, −9.040401288731661, −8.205034216347425, −7.892295998538264, −7.417176712275032, −6.922704288071553, −6.200490620289133, −5.835823813069527, −5.238836751051746, −4.600375230100977, −4.012586102136682, −3.583057629196827, −2.784400476368365, −2.507149257974599, −1.407419011007295, −0.8848854589184592, 0,
0.8848854589184592, 1.407419011007295, 2.507149257974599, 2.784400476368365, 3.583057629196827, 4.012586102136682, 4.600375230100977, 5.238836751051746, 5.835823813069527, 6.200490620289133, 6.922704288071553, 7.417176712275032, 7.892295998538264, 8.205034216347425, 9.040401288731661, 9.461969348782195, 9.780977750093730, 10.40699647735543, 10.94888792769853, 11.38246098475865, 12.07249880803757, 12.15574681653228, 12.72401735835964, 13.45441079664826, 13.76203882940716
