| L(s) = 1 | + 3·3-s + 5-s + 6·9-s − 3·11-s + 13-s + 3·15-s + 17-s − 4·19-s − 4·23-s + 25-s + 9·27-s + 9·29-s + 6·31-s − 9·33-s + 8·37-s + 3·39-s − 6·41-s − 8·43-s + 6·45-s − 7·47-s + 3·51-s + 8·53-s − 3·55-s − 12·57-s − 4·59-s + 10·61-s + 65-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 2·9-s − 0.904·11-s + 0.277·13-s + 0.774·15-s + 0.242·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s + 1.73·27-s + 1.67·29-s + 1.07·31-s − 1.56·33-s + 1.31·37-s + 0.480·39-s − 0.937·41-s − 1.21·43-s + 0.894·45-s − 1.02·47-s + 0.420·51-s + 1.09·53-s − 0.404·55-s − 1.58·57-s − 0.520·59-s + 1.28·61-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.880554341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.880554341\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−15.75219486830515, −15.37000841579756, −14.82643335082408, −14.31045309681131, −13.75212212520036, −13.35126341902157, −12.96598206660052, −12.23489006385360, −11.59856296545462, −10.56125299612288, −10.18368955519134, −9.781125113214317, −9.046183749226813, −8.367952733734979, −8.146745332814607, −7.608649007892710, −6.556665871232726, −6.338194288573618, −5.118782823436976, −4.608120953990888, −3.745644956356336, −3.146141939031800, −2.384424917747350, −2.013992912993971, −0.8693417682234287,
0.8693417682234287, 2.013992912993971, 2.384424917747350, 3.146141939031800, 3.745644956356336, 4.608120953990888, 5.118782823436976, 6.338194288573618, 6.556665871232726, 7.608649007892710, 8.146745332814607, 8.367952733734979, 9.046183749226813, 9.781125113214317, 10.18368955519134, 10.56125299612288, 11.59856296545462, 12.23489006385360, 12.96598206660052, 13.35126341902157, 13.75212212520036, 14.31045309681131, 14.82643335082408, 15.37000841579756, 15.75219486830515
