| L(s) = 1 | + 1.93·3-s + 3.34·5-s + 0.732·9-s − 11-s + 4.89·13-s + 6.46·15-s − 1.03·17-s + 6.69·19-s − 5.19·23-s + 6.19·25-s − 4.38·27-s − 4.92·29-s + 7.86·31-s − 1.93·33-s − 1.92·37-s + 9.46·39-s − 3.86·41-s + 4·43-s + 2.44·45-s + 6.03·47-s − 1.99·51-s − 10.9·53-s − 3.34·55-s + 12.9·57-s − 7.20·59-s + 14.4·61-s + 16.3·65-s + ⋯ |
| L(s) = 1 | + 1.11·3-s + 1.49·5-s + 0.244·9-s − 0.301·11-s + 1.35·13-s + 1.66·15-s − 0.251·17-s + 1.53·19-s − 1.08·23-s + 1.23·25-s − 0.843·27-s − 0.915·29-s + 1.41·31-s − 0.336·33-s − 0.316·37-s + 1.51·39-s − 0.603·41-s + 0.609·43-s + 0.365·45-s + 0.880·47-s − 0.280·51-s − 1.50·53-s − 0.451·55-s + 1.71·57-s − 0.938·59-s + 1.84·61-s + 2.03·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.567238981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.567238981\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 1.93T + 3T^{2} \) |
| 5 | \( 1 - 3.34T + 5T^{2} \) |
| 13 | \( 1 - 4.89T + 13T^{2} \) |
| 17 | \( 1 + 1.03T + 17T^{2} \) |
| 19 | \( 1 - 6.69T + 19T^{2} \) |
| 23 | \( 1 + 5.19T + 23T^{2} \) |
| 29 | \( 1 + 4.92T + 29T^{2} \) |
| 31 | \( 1 - 7.86T + 31T^{2} \) |
| 37 | \( 1 + 1.92T + 37T^{2} \) |
| 41 | \( 1 + 3.86T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 6.03T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 7.20T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 - 9.73T + 67T^{2} \) |
| 71 | \( 1 + 1.19T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 - 7.46T + 79T^{2} \) |
| 83 | \( 1 - 9.79T + 83T^{2} \) |
| 89 | \( 1 - 1.27T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−9.197348263682152405922421141752, −8.356959367587169834504668532152, −7.78221920126549892844255779738, −6.65474076445883935721772563693, −5.88567747881607786872481739749, −5.29299577256678742698779865751, −3.95496628925393867570445486162, −3.08192488319411278390065343511, −2.25388962863416925649715760761, −1.35162707997490870055865233575,
1.35162707997490870055865233575, 2.25388962863416925649715760761, 3.08192488319411278390065343511, 3.95496628925393867570445486162, 5.29299577256678742698779865751, 5.88567747881607786872481739749, 6.65474076445883935721772563693, 7.78221920126549892844255779738, 8.356959367587169834504668532152, 9.197348263682152405922421141752
