L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 2·6-s − 7-s + 9-s + 4·11-s − 2·12-s + 13-s + 2·14-s − 4·16-s − 4·17-s − 2·18-s + 5·19-s + 21-s − 8·22-s − 2·23-s − 2·26-s − 27-s − 2·28-s − 2·29-s − 5·31-s + 8·32-s − 4·33-s + 8·34-s + 2·36-s − 6·37-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.577·12-s + 0.277·13-s + 0.534·14-s − 16-s − 0.970·17-s − 0.471·18-s + 1.14·19-s + 0.218·21-s − 1.70·22-s − 0.417·23-s − 0.392·26-s − 0.192·27-s − 0.377·28-s − 0.371·29-s − 0.898·31-s + 1.41·32-s − 0.696·33-s + 1.37·34-s + 1/3·36-s − 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 47 | \( 1 + T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−8.373202677872493699098303003876, −7.45064721072443676979386133477, −6.88444334765109665029681185938, −6.27773173218796538808259578109, −5.31403028194459193974670492017, −4.30458566587860771229987474927, −3.44860078427357484618972072227, −2.01481495491150632302001767197, −1.17177156947527591608180583975, 0,
1.17177156947527591608180583975, 2.01481495491150632302001767197, 3.44860078427357484618972072227, 4.30458566587860771229987474927, 5.31403028194459193974670492017, 6.27773173218796538808259578109, 6.88444334765109665029681185938, 7.45064721072443676979386133477, 8.373202677872493699098303003876