| L(s) = 1 | − 5-s − 7-s + 2·13-s + 2·17-s − 7·19-s + 6·23-s + 25-s + 8·29-s + 3·31-s + 35-s + 37-s − 6·41-s − 4·47-s − 6·49-s − 10·53-s + 4·59-s + 11·61-s − 2·65-s + 5·67-s + 6·71-s − 73-s − 79-s + 12·83-s − 2·85-s − 10·89-s − 2·91-s + 7·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 0.554·13-s + 0.485·17-s − 1.60·19-s + 1.25·23-s + 1/5·25-s + 1.48·29-s + 0.538·31-s + 0.169·35-s + 0.164·37-s − 0.937·41-s − 0.583·47-s − 6/7·49-s − 1.37·53-s + 0.520·59-s + 1.40·61-s − 0.248·65-s + 0.610·67-s + 0.712·71-s − 0.117·73-s − 0.112·79-s + 1.31·83-s − 0.216·85-s − 1.05·89-s − 0.209·91-s + 0.718·95-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 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−14.25032343282412, −13.58257229844335, −13.03892715789425, −12.75844970894237, −12.23518197743735, −11.65671476337390, −11.15707120154689, −10.73848608813422, −10.16854376284582, −9.755487753166153, −9.059147913545878, −8.516308205485548, −8.218235788357796, −7.680769087318146, −6.742461284611259, −6.621985324586615, −6.148693292967483, −5.140766111828933, −4.927984622720780, −4.118918216805897, −3.653548983997505, −2.984624807546367, −2.484176739154353, −1.544259097987832, −0.8727718110836697, 0,
0.8727718110836697, 1.544259097987832, 2.484176739154353, 2.984624807546367, 3.653548983997505, 4.118918216805897, 4.927984622720780, 5.140766111828933, 6.148693292967483, 6.621985324586615, 6.742461284611259, 7.680769087318146, 8.218235788357796, 8.516308205485548, 9.059147913545878, 9.755487753166153, 10.16854376284582, 10.73848608813422, 11.15707120154689, 11.65671476337390, 12.23518197743735, 12.75844970894237, 13.03892715789425, 13.58257229844335, 14.25032343282412
