L(s) = 1 | − 2·2-s − 3-s + 2·4-s − 5-s + 2·6-s + 7-s + 9-s + 2·10-s − 2·11-s − 2·12-s + 13-s − 2·14-s + 15-s − 4·16-s − 4·17-s − 2·18-s + 3·19-s − 2·20-s − 21-s + 4·22-s − 9·23-s − 4·25-s − 2·26-s − 27-s + 2·28-s − 29-s − 2·30-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s + 0.377·7-s + 1/3·9-s + 0.632·10-s − 0.603·11-s − 0.577·12-s + 0.277·13-s − 0.534·14-s + 0.258·15-s − 16-s − 0.970·17-s − 0.471·18-s + 0.688·19-s − 0.447·20-s − 0.218·21-s + 0.852·22-s − 1.87·23-s − 4/5·25-s − 0.392·26-s − 0.192·27-s + 0.377·28-s − 0.185·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−11.24202046328869917288044685866, −10.40255846942711435084403033428, −9.599789620513966436808406248057, −8.454021307473714596721218552866, −7.77622984507062955719788276727, −6.81213071000907532665593378150, −5.44065418242704282313601393426, −4.07283187660253737667379106813, −1.90263358533576758230802068213, 0,
1.90263358533576758230802068213, 4.07283187660253737667379106813, 5.44065418242704282313601393426, 6.81213071000907532665593378150, 7.77622984507062955719788276727, 8.454021307473714596721218552866, 9.599789620513966436808406248057, 10.40255846942711435084403033428, 11.24202046328869917288044685866