| L(s) = 1 | − 3-s + 5-s + 2.73·7-s + 9-s − 4.73·11-s + 0.732·13-s − 15-s − 19-s − 2.73·21-s − 3.46·23-s + 25-s − 27-s + 8.19·29-s − 8.92·31-s + 4.73·33-s + 2.73·35-s − 6.19·37-s − 0.732·39-s + 1.26·41-s − 4.19·43-s + 45-s + 3.46·47-s + 0.464·49-s − 9.46·53-s − 4.73·55-s + 57-s − 2.53·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.03·7-s + 0.333·9-s − 1.42·11-s + 0.203·13-s − 0.258·15-s − 0.229·19-s − 0.596·21-s − 0.722·23-s + 0.200·25-s − 0.192·27-s + 1.52·29-s − 1.60·31-s + 0.823·33-s + 0.461·35-s − 1.01·37-s − 0.117·39-s + 0.198·41-s − 0.639·43-s + 0.149·45-s + 0.505·47-s + 0.0663·49-s − 1.29·53-s − 0.638·55-s + 0.132·57-s − 0.330·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 13 | \( 1 - 0.732T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 8.19T + 29T^{2} \) |
| 31 | \( 1 + 8.92T + 31T^{2} \) |
| 37 | \( 1 + 6.19T + 37T^{2} \) |
| 41 | \( 1 - 1.26T + 41T^{2} \) |
| 43 | \( 1 + 4.19T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + 9.46T + 53T^{2} \) |
| 59 | \( 1 + 2.53T + 59T^{2} \) |
| 61 | \( 1 + 6.53T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 4.39T + 71T^{2} \) |
| 73 | \( 1 + 16.9T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 6.19T + 97T^{2} \) |
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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
−7.950953049318770412735907912873, −7.32334924772435302710126925975, −6.39156163684891117821971290387, −5.67209228302257344780254716414, −5.03278579845189155581088427552, −4.50892939430143810019069048860, −3.31228133638644354575054727177, −2.24872077202480332877347288775, −1.45679254633042410870096967824, 0,
1.45679254633042410870096967824, 2.24872077202480332877347288775, 3.31228133638644354575054727177, 4.50892939430143810019069048860, 5.03278579845189155581088427552, 5.67209228302257344780254716414, 6.39156163684891117821971290387, 7.32334924772435302710126925975, 7.950953049318770412735907912873
