L(s) = 1 | + 3-s + 5-s − 2·7-s + 9-s − 4·11-s + 13-s + 15-s + 6·19-s − 2·21-s + 6·23-s + 25-s + 27-s − 8·29-s − 4·33-s − 2·35-s + 10·37-s + 39-s − 10·41-s + 12·43-s + 45-s + 12·47-s − 3·49-s − 6·53-s − 4·55-s + 6·57-s − 12·59-s − 10·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 0.258·15-s + 1.37·19-s − 0.436·21-s + 1.25·23-s + 1/5·25-s + 0.192·27-s − 1.48·29-s − 0.696·33-s − 0.338·35-s + 1.64·37-s + 0.160·39-s − 1.56·41-s + 1.82·43-s + 0.149·45-s + 1.75·47-s − 3/7·49-s − 0.824·53-s − 0.539·55-s + 0.794·57-s − 1.56·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.376073549\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.376073549\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.76310261646238316815587032045, −7.60617401651616499744205466471, −6.68617479432368318865338085693, −5.85007123338141562167207145117, −5.28688824712567302978364533670, −4.41852579022327940408754332822, −3.29523173754832967510828137543, −2.94226252265055894012921395295, −1.98199092536765963391288078697, −0.77552881284955596024008457791,
0.77552881284955596024008457791, 1.98199092536765963391288078697, 2.94226252265055894012921395295, 3.29523173754832967510828137543, 4.41852579022327940408754332822, 5.28688824712567302978364533670, 5.85007123338141562167207145117, 6.68617479432368318865338085693, 7.60617401651616499744205466471, 7.76310261646238316815587032045