L(s) = 1 | − 3-s + 3·5-s − 2·7-s + 9-s − 6·11-s − 3·15-s − 3·17-s − 2·19-s + 2·21-s + 6·23-s + 4·25-s − 27-s + 3·29-s + 4·31-s + 6·33-s − 6·35-s − 7·37-s − 3·41-s + 10·43-s + 3·45-s − 6·47-s − 3·49-s + 3·51-s + 3·53-s − 18·55-s + 2·57-s − 7·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.34·5-s − 0.755·7-s + 1/3·9-s − 1.80·11-s − 0.774·15-s − 0.727·17-s − 0.458·19-s + 0.436·21-s + 1.25·23-s + 4/5·25-s − 0.192·27-s + 0.557·29-s + 0.718·31-s + 1.04·33-s − 1.01·35-s − 1.15·37-s − 0.468·41-s + 1.52·43-s + 0.447·45-s − 0.875·47-s − 3/7·49-s + 0.420·51-s + 0.412·53-s − 2.42·55-s + 0.264·57-s − 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.353557831\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.353557831\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−7.71228890286374164796051264470, −6.94066222555462250579818106752, −6.33681695871009114332207145716, −5.81439251540901211398105316257, −5.06733563554819733912021363064, −4.65047702520769116257093010231, −3.28311515952908152854941075889, −2.59687322896259811851599454306, −1.87740828212772298640276912541, −0.56719806990473160362827845912,
0.56719806990473160362827845912, 1.87740828212772298640276912541, 2.59687322896259811851599454306, 3.28311515952908152854941075889, 4.65047702520769116257093010231, 5.06733563554819733912021363064, 5.81439251540901211398105316257, 6.33681695871009114332207145716, 6.94066222555462250579818106752, 7.71228890286374164796051264470