L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 5-s + 2·6-s + 2·7-s + 9-s − 2·10-s − 5·11-s − 2·12-s − 13-s − 4·14-s − 15-s − 4·16-s + 4·17-s − 2·18-s − 2·19-s + 2·20-s − 2·21-s + 10·22-s − 2·23-s + 25-s + 2·26-s − 27-s + 4·28-s + 2·30-s + 31-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s + 0.755·7-s + 1/3·9-s − 0.632·10-s − 1.50·11-s − 0.577·12-s − 0.277·13-s − 1.06·14-s − 0.258·15-s − 16-s + 0.970·17-s − 0.471·18-s − 0.458·19-s + 0.447·20-s − 0.436·21-s + 2.13·22-s − 0.417·23-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.755·28-s + 0.365·30-s + 0.179·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.71817622620555514083532956189, −7.43271862759958102308614169157, −6.42857313216203754437592947734, −5.59353945208036209896218077278, −5.02701575305336396584728078178, −4.22619977045172942057501483258, −2.78534919798884151165950607258, −2.00167922815022247928729579925, −1.09631698869590547793298641274, 0,
1.09631698869590547793298641274, 2.00167922815022247928729579925, 2.78534919798884151165950607258, 4.22619977045172942057501483258, 5.02701575305336396584728078178, 5.59353945208036209896218077278, 6.42857313216203754437592947734, 7.43271862759958102308614169157, 7.71817622620555514083532956189