| L(s) = 1 | + 5-s + 2·7-s + 11-s − 17-s − 4·23-s + 25-s − 2·29-s − 4·31-s + 2·35-s − 2·37-s + 6·43-s − 3·49-s + 6·53-s + 55-s + 14·59-s − 2·61-s − 14·67-s − 2·71-s + 16·73-s + 2·77-s − 12·79-s − 85-s + 18·89-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s + 0.301·11-s − 0.242·17-s − 0.834·23-s + 1/5·25-s − 0.371·29-s − 0.718·31-s + 0.338·35-s − 0.328·37-s + 0.914·43-s − 3/7·49-s + 0.824·53-s + 0.134·55-s + 1.82·59-s − 0.256·61-s − 1.71·67-s − 0.237·71-s + 1.87·73-s + 0.227·77-s − 1.35·79-s − 0.108·85-s + 1.90·89-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−13.67513426901731, −13.29899814589354, −12.71329991725044, −12.27766499215631, −11.65738618597528, −11.39060343703134, −10.77664023758001, −10.34078327217848, −9.875153961121361, −9.235852635805091, −8.895817481026905, −8.341452773823202, −7.826333416997248, −7.282930198647393, −6.858589451914842, −6.020584557295463, −5.889896533449271, −5.047496295897169, −4.784431216918810, −3.893308312981187, −3.699129173602159, −2.693916714896527, −2.176187743157044, −1.632671986224957, −0.9591276634314386, 0,
0.9591276634314386, 1.632671986224957, 2.176187743157044, 2.693916714896527, 3.699129173602159, 3.893308312981187, 4.784431216918810, 5.047496295897169, 5.889896533449271, 6.020584557295463, 6.858589451914842, 7.282930198647393, 7.826333416997248, 8.341452773823202, 8.895817481026905, 9.235852635805091, 9.875153961121361, 10.34078327217848, 10.77664023758001, 11.39060343703134, 11.65738618597528, 12.27766499215631, 12.71329991725044, 13.29899814589354, 13.67513426901731
