| L(s) = 1 | + 3-s + 0.732·5-s + 7-s + 9-s − 4·11-s − 1.46·13-s + 0.732·15-s − 3.26·17-s + 19-s + 21-s − 6.92·23-s − 4.46·25-s + 27-s − 3.26·29-s − 2·31-s − 4·33-s + 0.732·35-s − 4.92·37-s − 1.46·39-s − 8.92·41-s + 4.92·43-s + 0.732·45-s + 0.196·47-s + 49-s − 3.26·51-s + 7.66·53-s − 2.92·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.327·5-s + 0.377·7-s + 0.333·9-s − 1.20·11-s − 0.406·13-s + 0.189·15-s − 0.792·17-s + 0.229·19-s + 0.218·21-s − 1.44·23-s − 0.892·25-s + 0.192·27-s − 0.606·29-s − 0.359·31-s − 0.696·33-s + 0.123·35-s − 0.810·37-s − 0.234·39-s − 1.39·41-s + 0.751·43-s + 0.109·45-s + 0.0286·47-s + 0.142·49-s − 0.457·51-s + 1.05·53-s − 0.394·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 0.732T + 5T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 + 3.26T + 17T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 + 3.26T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 4.92T + 37T^{2} \) |
| 41 | \( 1 + 8.92T + 41T^{2} \) |
| 43 | \( 1 - 4.92T + 43T^{2} \) |
| 47 | \( 1 - 0.196T + 47T^{2} \) |
| 53 | \( 1 - 7.66T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 - 0.928T + 61T^{2} \) |
| 67 | \( 1 - 5.46T + 67T^{2} \) |
| 71 | \( 1 - 4.19T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 - 2.92T + 79T^{2} \) |
| 83 | \( 1 - 8.19T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−8.215080694543461831295586226946, −7.69077403006010244619603218209, −6.96597955757863350195984661075, −5.91836790843344527318839669470, −5.24732949074927538314814056862, −4.38183196384062589024908734169, −3.48395451475226006682558209041, −2.38738853561610578284772266510, −1.83437597084645015196398689254, 0,
1.83437597084645015196398689254, 2.38738853561610578284772266510, 3.48395451475226006682558209041, 4.38183196384062589024908734169, 5.24732949074927538314814056862, 5.91836790843344527318839669470, 6.96597955757863350195984661075, 7.69077403006010244619603218209, 8.215080694543461831295586226946
