| L(s) = 1 | + 2·3-s + 16·5-s + 7·7-s − 23·9-s + 8·11-s + 28·13-s + 32·15-s + 54·17-s + 110·19-s + 14·21-s − 48·23-s + 131·25-s − 100·27-s − 110·29-s − 12·31-s + 16·33-s + 112·35-s − 246·37-s + 56·39-s + 182·41-s − 128·43-s − 368·45-s − 324·47-s + 49·49-s + 108·51-s − 162·53-s + 128·55-s + ⋯ |
| L(s) = 1 | + 0.384·3-s + 1.43·5-s + 0.377·7-s − 0.851·9-s + 0.219·11-s + 0.597·13-s + 0.550·15-s + 0.770·17-s + 1.32·19-s + 0.145·21-s − 0.435·23-s + 1.04·25-s − 0.712·27-s − 0.704·29-s − 0.0695·31-s + 0.0844·33-s + 0.540·35-s − 1.09·37-s + 0.229·39-s + 0.693·41-s − 0.453·43-s − 1.21·45-s − 1.00·47-s + 1/7·49-s + 0.296·51-s − 0.419·53-s + 0.313·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.238790517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.238790517\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 5 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 8 T + p^{3} T^{2} \) |
| 13 | \( 1 - 28 T + p^{3} T^{2} \) |
| 17 | \( 1 - 54 T + p^{3} T^{2} \) |
| 19 | \( 1 - 110 T + p^{3} T^{2} \) |
| 23 | \( 1 + 48 T + p^{3} T^{2} \) |
| 29 | \( 1 + 110 T + p^{3} T^{2} \) |
| 31 | \( 1 + 12 T + p^{3} T^{2} \) |
| 37 | \( 1 + 246 T + p^{3} T^{2} \) |
| 41 | \( 1 - 182 T + p^{3} T^{2} \) |
| 43 | \( 1 + 128 T + p^{3} T^{2} \) |
| 47 | \( 1 + 324 T + p^{3} T^{2} \) |
| 53 | \( 1 + 162 T + p^{3} T^{2} \) |
| 59 | \( 1 + 810 T + p^{3} T^{2} \) |
| 61 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 67 | \( 1 + 244 T + p^{3} T^{2} \) |
| 71 | \( 1 - 768 T + p^{3} T^{2} \) |
| 73 | \( 1 + 702 T + p^{3} T^{2} \) |
| 79 | \( 1 + 440 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1302 T + p^{3} T^{2} \) |
| 89 | \( 1 - 730 T + p^{3} T^{2} \) |
| 97 | \( 1 - 294 T + p^{3} T^{2} \) |
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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
−13.54117189668371036011703847370, −12.12751027932725195655808190551, −10.98196705529977912761465782264, −9.783767419262316707766559141748, −8.997639643336176546624282026017, −7.77007301686034376969002054558, −6.14825372394142740241001389312, −5.29322772507919995748071234489, −3.21965443014692344273688757788, −1.65531954476772920067548415280,
1.65531954476772920067548415280, 3.21965443014692344273688757788, 5.29322772507919995748071234489, 6.14825372394142740241001389312, 7.77007301686034376969002054558, 8.997639643336176546624282026017, 9.783767419262316707766559141748, 10.98196705529977912761465782264, 12.12751027932725195655808190551, 13.54117189668371036011703847370
