| L(s) = 1 | + 5-s + 7-s + 6·11-s + 2·13-s + 4·19-s + 9·23-s + 25-s − 3·29-s + 4·31-s + 35-s + 8·37-s + 3·41-s − 8·43-s − 3·47-s − 6·49-s − 6·53-s + 6·55-s + 6·59-s − 13·61-s + 2·65-s + 13·67-s − 6·71-s − 4·73-s + 6·77-s + 10·79-s − 9·83-s − 9·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 1.80·11-s + 0.554·13-s + 0.917·19-s + 1.87·23-s + 1/5·25-s − 0.557·29-s + 0.718·31-s + 0.169·35-s + 1.31·37-s + 0.468·41-s − 1.21·43-s − 0.437·47-s − 6/7·49-s − 0.824·53-s + 0.809·55-s + 0.781·59-s − 1.66·61-s + 0.248·65-s + 1.58·67-s − 0.712·71-s − 0.468·73-s + 0.683·77-s + 1.12·79-s − 0.987·83-s − 0.953·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.109995706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.109995706\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−8.094441013633697164904296060915, −7.16987750958405593706086339273, −6.60178256262327875796890769260, −6.00241662996362181901419988914, −5.12527704701765855134543949312, −4.45049154688339943623164827018, −3.57034236623571612011571855945, −2.84852147358821991378367734431, −1.55340894367064703071797939365, −1.06146439917456403229786028604,
1.06146439917456403229786028604, 1.55340894367064703071797939365, 2.84852147358821991378367734431, 3.57034236623571612011571855945, 4.45049154688339943623164827018, 5.12527704701765855134543949312, 6.00241662996362181901419988914, 6.60178256262327875796890769260, 7.16987750958405593706086339273, 8.094441013633697164904296060915
