| L(s) = 1 | − 1.41·2-s − 6.00·4-s + 6.16·5-s − 0.219·7-s + 19.7·8-s − 8.71·10-s − 52.4·11-s + 86.8·13-s + 0.310·14-s + 20.0·16-s − 94.7·17-s − 41.3·19-s − 37.0·20-s + 74.1·22-s + 23·23-s − 86.9·25-s − 122.·26-s + 1.31·28-s − 221.·29-s − 86.5·31-s − 186.·32-s + 133.·34-s − 1.35·35-s − 363.·37-s + 58.4·38-s + 122.·40-s − 342.·41-s + ⋯ |
| L(s) = 1 | − 0.499·2-s − 0.750·4-s + 0.551·5-s − 0.0118·7-s + 0.874·8-s − 0.275·10-s − 1.43·11-s + 1.85·13-s + 0.00593·14-s + 0.313·16-s − 1.35·17-s − 0.499·19-s − 0.413·20-s + 0.718·22-s + 0.208·23-s − 0.695·25-s − 0.925·26-s + 0.00890·28-s − 1.42·29-s − 0.501·31-s − 1.03·32-s + 0.675·34-s − 0.00654·35-s − 1.61·37-s + 0.249·38-s + 0.482·40-s − 1.30·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 + 1.41T + 8T^{2} \) |
| 5 | \( 1 - 6.16T + 125T^{2} \) |
| 7 | \( 1 + 0.219T + 343T^{2} \) |
| 11 | \( 1 + 52.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 86.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 94.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 41.3T + 6.85e3T^{2} \) |
| 29 | \( 1 + 221.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 86.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 363.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 342.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 454.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 442.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 456.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 753.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 75.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 551.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 565.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 573.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 37.6T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.16e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 244.T + 9.12e5T^{2} \) |
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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
−10.96283253826282592507687941552, −10.58701811744454290712851437980, −9.300280336992913736468306474638, −8.638015596908476045544607769014, −7.65209701437963846352751321792, −6.16113613068095045490263585361, −5.09641257539077605055858205497, −3.74377437994630710037391279003, −1.85008788955255684137281809263, 0,
1.85008788955255684137281809263, 3.74377437994630710037391279003, 5.09641257539077605055858205497, 6.16113613068095045490263585361, 7.65209701437963846352751321792, 8.638015596908476045544607769014, 9.300280336992913736468306474638, 10.58701811744454290712851437980, 10.96283253826282592507687941552
