| L(s) = 1 | + 2.13·2-s + 5.69·3-s − 3.42·4-s + 17.0·5-s + 12.1·6-s − 24.4·8-s + 5.41·9-s + 36.4·10-s + 19.6·11-s − 19.5·12-s + 63.2·13-s + 96.9·15-s − 24.8·16-s + 78.4·17-s + 11.5·18-s − 11.2·19-s − 58.3·20-s + 41.9·22-s + 89.8·23-s − 139.·24-s + 164.·25-s + 135.·26-s − 122.·27-s + 45.5·29-s + 207.·30-s − 79.1·31-s + 142.·32-s + ⋯ |
| L(s) = 1 | + 0.755·2-s + 1.09·3-s − 0.428·4-s + 1.52·5-s + 0.828·6-s − 1.07·8-s + 0.200·9-s + 1.15·10-s + 0.537·11-s − 0.469·12-s + 1.35·13-s + 1.66·15-s − 0.387·16-s + 1.11·17-s + 0.151·18-s − 0.136·19-s − 0.652·20-s + 0.406·22-s + 0.814·23-s − 1.18·24-s + 1.31·25-s + 1.02·26-s − 0.875·27-s + 0.291·29-s + 1.26·30-s − 0.458·31-s + 0.786·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.568183770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.568183770\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 47 | \( 1 - 47T \) |
| good | 2 | \( 1 - 2.13T + 8T^{2} \) |
| 3 | \( 1 - 5.69T + 27T^{2} \) |
| 5 | \( 1 - 17.0T + 125T^{2} \) |
| 11 | \( 1 - 19.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 63.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 78.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 11.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 89.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 45.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 79.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 176.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 9.06T + 6.89e4T^{2} \) |
| 43 | \( 1 - 187.T + 7.95e4T^{2} \) |
| 53 | \( 1 - 183.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 368.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 82.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 87.5T + 3.00e5T^{2} \) |
| 71 | \( 1 - 28.0T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 350.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.02e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 564.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 479.T + 9.12e5T^{2} \) |
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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
−8.924519868741786967788511097883, −8.137465943497064138064617294668, −6.96196694157079784161715903985, −5.89692531674224960874299495166, −5.72549613619264725105561171595, −4.60932437597053714429909725177, −3.54389689369594941218754821302, −3.08979924083573621953724998891, −2.00448999137414237321931638349, −1.03988412615540981110206644781,
1.03988412615540981110206644781, 2.00448999137414237321931638349, 3.08979924083573621953724998891, 3.54389689369594941218754821302, 4.60932437597053714429909725177, 5.72549613619264725105561171595, 5.89692531674224960874299495166, 6.96196694157079784161715903985, 8.137465943497064138064617294668, 8.924519868741786967788511097883
