| L(s) = 1 | + 2-s − 3·3-s − 7·4-s + 5·5-s − 3·6-s + 7·7-s − 15·8-s + 9·9-s + 5·10-s + 50·11-s + 21·12-s − 28·13-s + 7·14-s − 15·15-s + 41·16-s − 22·17-s + 9·18-s − 84·19-s − 35·20-s − 21·21-s + 50·22-s + 23·23-s + 45·24-s + 25·25-s − 28·26-s − 27·27-s − 49·28-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 0.577·3-s − 7/8·4-s + 0.447·5-s − 0.204·6-s + 0.377·7-s − 0.662·8-s + 1/3·9-s + 0.158·10-s + 1.37·11-s + 0.505·12-s − 0.597·13-s + 0.133·14-s − 0.258·15-s + 0.640·16-s − 0.313·17-s + 0.117·18-s − 1.01·19-s − 0.391·20-s − 0.218·21-s + 0.484·22-s + 0.208·23-s + 0.382·24-s + 1/5·25-s − 0.211·26-s − 0.192·27-s − 0.330·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| 23 | \( 1 - p T \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 11 | \( 1 - 50 T + p^{3} T^{2} \) |
| 13 | \( 1 + 28 T + p^{3} T^{2} \) |
| 17 | \( 1 + 22 T + p^{3} T^{2} \) |
| 19 | \( 1 + 84 T + p^{3} T^{2} \) |
| 29 | \( 1 + 24 T + p^{3} T^{2} \) |
| 31 | \( 1 + 140 T + p^{3} T^{2} \) |
| 37 | \( 1 + 170 T + p^{3} T^{2} \) |
| 41 | \( 1 - 302 T + p^{3} T^{2} \) |
| 43 | \( 1 + 194 T + p^{3} T^{2} \) |
| 47 | \( 1 - 56 T + p^{3} T^{2} \) |
| 53 | \( 1 + 30 T + p^{3} T^{2} \) |
| 59 | \( 1 - 784 T + p^{3} T^{2} \) |
| 61 | \( 1 + 326 T + p^{3} T^{2} \) |
| 67 | \( 1 - 918 T + p^{3} T^{2} \) |
| 71 | \( 1 + 402 T + p^{3} T^{2} \) |
| 73 | \( 1 - 38 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1112 T + p^{3} T^{2} \) |
| 83 | \( 1 - 132 T + p^{3} T^{2} \) |
| 89 | \( 1 - 636 T + p^{3} T^{2} \) |
| 97 | \( 1 - 448 T + p^{3} T^{2} \) |
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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
−8.439164855889793070063815914154, −7.28244187610096763980723294429, −6.51309469646935983548147710586, −5.78410555501087558776731796301, −5.01539470041070997028763559807, −4.31098167112310865077000586114, −3.61396038055922282367723182490, −2.20616563019957144251804867475, −1.13670458465540043245549522748, 0,
1.13670458465540043245549522748, 2.20616563019957144251804867475, 3.61396038055922282367723182490, 4.31098167112310865077000586114, 5.01539470041070997028763559807, 5.78410555501087558776731796301, 6.51309469646935983548147710586, 7.28244187610096763980723294429, 8.439164855889793070063815914154
