| L(s) = 1 | − 2·2-s + 5.78·3-s + 4·4-s + 6.19·5-s − 11.5·6-s + 11.2·7-s − 8·8-s + 6.45·9-s − 12.3·10-s − 45.1·11-s + 23.1·12-s − 22.6·13-s − 22.4·14-s + 35.8·15-s + 16·16-s − 31.0·17-s − 12.9·18-s + 24.7·20-s + 64.9·21-s + 90.2·22-s − 144.·23-s − 46.2·24-s − 86.5·25-s + 45.3·26-s − 118.·27-s + 44.8·28-s − 24.2·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.11·3-s + 0.5·4-s + 0.554·5-s − 0.787·6-s + 0.605·7-s − 0.353·8-s + 0.238·9-s − 0.392·10-s − 1.23·11-s + 0.556·12-s − 0.483·13-s − 0.428·14-s + 0.617·15-s + 0.250·16-s − 0.442·17-s − 0.168·18-s + 0.277·20-s + 0.674·21-s + 0.874·22-s − 1.30·23-s − 0.393·24-s − 0.692·25-s + 0.341·26-s − 0.847·27-s + 0.302·28-s − 0.155·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 5.78T + 27T^{2} \) |
| 5 | \( 1 - 6.19T + 125T^{2} \) |
| 7 | \( 1 - 11.2T + 343T^{2} \) |
| 11 | \( 1 + 45.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 22.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 31.0T + 4.91e3T^{2} \) |
| 23 | \( 1 + 144.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 24.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 303.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 121.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 103.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 444.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 481.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 441.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 242.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 531.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 727.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 225.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 351.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 602.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 248.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.66e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 846.T + 9.12e5T^{2} \) |
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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
−9.474046534601007744889899648784, −8.731766590934643853792420757853, −7.87927863014741198040470698554, −7.49188707510592526501614275596, −6.06137664394079752379738008552, −5.12896643138669427056932488677, −3.70057948436342476589095431467, −2.41421961935403993086814589815, −1.93149489813974733415692701622, 0,
1.93149489813974733415692701622, 2.41421961935403993086814589815, 3.70057948436342476589095431467, 5.12896643138669427056932488677, 6.06137664394079752379738008552, 7.49188707510592526501614275596, 7.87927863014741198040470698554, 8.731766590934643853792420757853, 9.474046534601007744889899648784
