| L(s) = 1 | + 3-s + 4·7-s + 9-s − 4·11-s − 2·13-s + 6·17-s + 4·19-s + 4·21-s + 27-s − 2·29-s + 4·31-s − 4·33-s − 2·37-s − 2·39-s + 2·41-s + 4·43-s − 8·47-s + 9·49-s + 6·51-s + 10·53-s + 4·57-s + 4·59-s − 6·61-s + 4·63-s + 4·67-s − 16·71-s + 6·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 1.45·17-s + 0.917·19-s + 0.872·21-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.696·33-s − 0.328·37-s − 0.320·39-s + 0.312·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s + 0.840·51-s + 1.37·53-s + 0.529·57-s + 0.520·59-s − 0.768·61-s + 0.503·63-s + 0.488·67-s − 1.89·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.963484450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.963484450\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.068910377622674576022530973520, −7.70532577155240774567270034344, −7.25082360725771695647207513608, −5.92492560055636915788477830174, −5.14012488832414091996399426491, −4.79714294650582000191829067487, −3.66630001779805216750222801300, −2.80556018209453912258899233087, −1.98560467024152423573604385963, −0.968519021125122067277353981577,
0.968519021125122067277353981577, 1.98560467024152423573604385963, 2.80556018209453912258899233087, 3.66630001779805216750222801300, 4.79714294650582000191829067487, 5.14012488832414091996399426491, 5.92492560055636915788477830174, 7.25082360725771695647207513608, 7.70532577155240774567270034344, 8.068910377622674576022530973520
