L(s) = 1 | + 3-s + 4·5-s + 3·7-s − 2·9-s + 5·11-s − 13-s + 4·15-s + 8·19-s + 3·21-s − 9·23-s + 11·25-s − 5·27-s + 6·29-s + 2·31-s + 5·33-s + 12·35-s − 2·37-s − 39-s − 10·41-s + 5·43-s − 8·45-s + 47-s + 2·49-s + 6·53-s + 20·55-s + 8·57-s + 4·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.78·5-s + 1.13·7-s − 2/3·9-s + 1.50·11-s − 0.277·13-s + 1.03·15-s + 1.83·19-s + 0.654·21-s − 1.87·23-s + 11/5·25-s − 0.962·27-s + 1.11·29-s + 0.359·31-s + 0.870·33-s + 2.02·35-s − 0.328·37-s − 0.160·39-s − 1.56·41-s + 0.762·43-s − 1.19·45-s + 0.145·47-s + 2/7·49-s + 0.824·53-s + 2.69·55-s + 1.05·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.050414631\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.050414631\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.64753877339685, −14.33940707330135, −14.03658271125386, −13.80503596385815, −13.25116930992537, −12.20764733837353, −11.87928746082983, −11.50293923188989, −10.67889348542944, −9.964248429434269, −9.741879267363860, −9.140832536122021, −8.597096434985770, −8.173147110109037, −7.418742773352598, −6.697190641722233, −6.151902854580463, −5.546583316229765, −5.152191140431104, −4.361306546879449, −3.563922047816846, −2.822138968941326, −2.110181848037567, −1.622989555962023, −0.9602182236189895,
0.9602182236189895, 1.622989555962023, 2.110181848037567, 2.822138968941326, 3.563922047816846, 4.361306546879449, 5.152191140431104, 5.546583316229765, 6.151902854580463, 6.697190641722233, 7.418742773352598, 8.173147110109037, 8.597096434985770, 9.140832536122021, 9.741879267363860, 9.964248429434269, 10.67889348542944, 11.50293923188989, 11.87928746082983, 12.20764733837353, 13.25116930992537, 13.80503596385815, 14.03658271125386, 14.33940707330135, 14.64753877339685