L(s) = 1 | + 5-s − 2·7-s + 2·13-s + 2·17-s − 4·19-s − 2·23-s + 25-s − 2·29-s − 8·31-s − 2·35-s + 6·37-s + 10·41-s − 8·43-s − 6·47-s − 3·49-s − 2·53-s + 12·59-s − 2·61-s + 2·65-s + 4·67-s + 6·71-s + 10·79-s − 4·83-s + 2·85-s + 8·89-s − 4·91-s − 4·95-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.755·7-s + 0.554·13-s + 0.485·17-s − 0.917·19-s − 0.417·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s − 0.338·35-s + 0.986·37-s + 1.56·41-s − 1.21·43-s − 0.875·47-s − 3/7·49-s − 0.274·53-s + 1.56·59-s − 0.256·61-s + 0.248·65-s + 0.488·67-s + 0.712·71-s + 1.12·79-s − 0.439·83-s + 0.216·85-s + 0.847·89-s − 0.419·91-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−14.27852940892926, −13.50520371447765, −13.10173820473664, −12.77224185127140, −12.38748896985988, −11.55230428494557, −11.19169141482737, −10.69512629051556, −10.03281690565772, −9.749745941177040, −9.165966262168959, −8.741370999385792, −8.052593953958594, −7.641122285729947, −6.917182664093456, −6.337064441697074, −6.109043341965174, −5.404389370452147, −4.909542818476783, −4.013447868549162, −3.690253797609631, −3.011183273512010, −2.282873535622824, −1.737727700463738, −0.8648801991010523, 0,
0.8648801991010523, 1.737727700463738, 2.282873535622824, 3.011183273512010, 3.690253797609631, 4.013447868549162, 4.909542818476783, 5.404389370452147, 6.109043341965174, 6.337064441697074, 6.917182664093456, 7.641122285729947, 8.052593953958594, 8.741370999385792, 9.165966262168959, 9.749745941177040, 10.03281690565772, 10.69512629051556, 11.19169141482737, 11.55230428494557, 12.38748896985988, 12.77224185127140, 13.10173820473664, 13.50520371447765, 14.27852940892926