| L(s) = 1 | + 2-s + 3-s + 4-s + 3·5-s + 6-s + 7-s + 8-s − 2·9-s + 3·10-s + 12-s − 5·13-s + 14-s + 3·15-s + 16-s − 2·18-s − 2·19-s + 3·20-s + 21-s − 6·23-s + 24-s + 4·25-s − 5·26-s − 5·27-s + 28-s + 3·30-s − 4·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.948·10-s + 0.288·12-s − 1.38·13-s + 0.267·14-s + 0.774·15-s + 1/4·16-s − 0.471·18-s − 0.458·19-s + 0.670·20-s + 0.218·21-s − 1.25·23-s + 0.204·24-s + 4/5·25-s − 0.980·26-s − 0.962·27-s + 0.188·28-s + 0.547·30-s − 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19118 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19118 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−15.97698604639099, −15.07487641606685, −14.69300317351631, −14.35589540109062, −13.93765935449631, −13.35492540548392, −12.86027100994656, −12.29275815252612, −11.61812525061235, −11.14319782870733, −10.24785916634678, −9.968301440749515, −9.274775316656018, −8.792834560931701, −7.894192970363509, −7.578957114556116, −6.672545471839612, −5.961683017723770, −5.701546114134340, −4.860906609419769, −4.388379026718890, −3.366866811849367, −2.707656373535749, −2.105873589125401, −1.642147651115657, 0,
1.642147651115657, 2.105873589125401, 2.707656373535749, 3.366866811849367, 4.388379026718890, 4.860906609419769, 5.701546114134340, 5.961683017723770, 6.672545471839612, 7.578957114556116, 7.894192970363509, 8.792834560931701, 9.274775316656018, 9.968301440749515, 10.24785916634678, 11.14319782870733, 11.61812525061235, 12.29275815252612, 12.86027100994656, 13.35492540548392, 13.93765935449631, 14.35589540109062, 14.69300317351631, 15.07487641606685, 15.97698604639099
