L(s) = 1 | + 3-s − 4·7-s + 9-s + 13-s − 3·19-s − 4·21-s + 4·23-s + 27-s − 29-s − 8·31-s + 37-s + 39-s + 41-s − 6·43-s + 11·47-s + 9·49-s + 3·53-s − 3·57-s + 10·59-s + 4·61-s − 4·63-s − 13·67-s + 4·69-s + 9·71-s − 3·79-s + 81-s − 2·83-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.277·13-s − 0.688·19-s − 0.872·21-s + 0.834·23-s + 0.192·27-s − 0.185·29-s − 1.43·31-s + 0.164·37-s + 0.160·39-s + 0.156·41-s − 0.914·43-s + 1.60·47-s + 9/7·49-s + 0.412·53-s − 0.397·57-s + 1.30·59-s + 0.512·61-s − 0.503·63-s − 1.58·67-s + 0.481·69-s + 1.06·71-s − 0.337·79-s + 1/9·81-s − 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.41443855701677, −14.72698899220208, −14.47383557443083, −13.53588958723562, −13.25538113848622, −12.87504555534292, −12.33278957242244, −11.72458411111291, −10.92314633301433, −10.49778025675717, −9.931663567339949, −9.302284622457139, −8.983870362548026, −8.437807869100230, −7.641760148682328, −7.028258363888981, −6.681019757956649, −5.934615299336914, −5.425711832304923, −4.514580035720354, −3.767302117413220, −3.421058723609902, −2.660716029643702, −2.048569261014872, −0.9758961575919157, 0,
0.9758961575919157, 2.048569261014872, 2.660716029643702, 3.421058723609902, 3.767302117413220, 4.514580035720354, 5.425711832304923, 5.934615299336914, 6.681019757956649, 7.028258363888981, 7.641760148682328, 8.437807869100230, 8.983870362548026, 9.302284622457139, 9.931663567339949, 10.49778025675717, 10.92314633301433, 11.72458411111291, 12.33278957242244, 12.87504555534292, 13.25538113848622, 13.53588958723562, 14.47383557443083, 14.72698899220208, 15.41443855701677