L(s) = 1 | − 3-s + 2·7-s + 9-s + 11-s + 2·13-s − 4·17-s − 6·19-s − 2·21-s − 27-s − 8·29-s − 8·31-s − 33-s − 10·37-s − 2·39-s + 8·41-s + 2·43-s + 8·47-s − 3·49-s + 4·51-s + 2·53-s + 6·57-s + 12·59-s + 10·61-s + 2·63-s − 12·67-s + 8·71-s − 6·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.970·17-s − 1.37·19-s − 0.436·21-s − 0.192·27-s − 1.48·29-s − 1.43·31-s − 0.174·33-s − 1.64·37-s − 0.320·39-s + 1.24·41-s + 0.304·43-s + 1.16·47-s − 3/7·49-s + 0.560·51-s + 0.274·53-s + 0.794·57-s + 1.56·59-s + 1.28·61-s + 0.251·63-s − 1.46·67-s + 0.949·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.440430784025312372176404880920, −7.33836150250731100823677708924, −6.84554982340368952146699504752, −5.87235410191026440553599701768, −5.34944646633716512717399552762, −4.27326531503598949365280226332, −3.84596150576596792333851680234, −2.30690841830688128950640609404, −1.50952817403596062415232514980, 0,
1.50952817403596062415232514980, 2.30690841830688128950640609404, 3.84596150576596792333851680234, 4.27326531503598949365280226332, 5.34944646633716512717399552762, 5.87235410191026440553599701768, 6.84554982340368952146699504752, 7.33836150250731100823677708924, 8.440430784025312372176404880920