L(s) = 1 | + 3-s − 2·4-s − 7-s + 9-s − 11-s − 2·12-s − 13-s + 4·16-s − 17-s − 4·19-s − 21-s − 3·23-s + 27-s + 2·28-s − 8·29-s − 4·31-s − 33-s − 2·36-s + 3·37-s − 39-s − 9·41-s − 8·43-s + 2·44-s + 10·47-s + 4·48-s − 6·49-s − 51-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.577·12-s − 0.277·13-s + 16-s − 0.242·17-s − 0.917·19-s − 0.218·21-s − 0.625·23-s + 0.192·27-s + 0.377·28-s − 1.48·29-s − 0.718·31-s − 0.174·33-s − 1/3·36-s + 0.493·37-s − 0.160·39-s − 1.40·41-s − 1.21·43-s + 0.301·44-s + 1.45·47-s + 0.577·48-s − 6/7·49-s − 0.140·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 15 T + p T^{2} \) |
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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
−9.532862510891535243606980619416, −8.790775602793191528061887163246, −8.095863304798581886113130196548, −7.22491697018544276884142731851, −6.09052611465894365795243669910, −5.09006707574467841333534810174, −4.13458107803479074584674344701, −3.31456654216549001444250807099, −1.94623851103131791750784419509, 0,
1.94623851103131791750784419509, 3.31456654216549001444250807099, 4.13458107803479074584674344701, 5.09006707574467841333534810174, 6.09052611465894365795243669910, 7.22491697018544276884142731851, 8.095863304798581886113130196548, 8.790775602793191528061887163246, 9.532862510891535243606980619416