L(s) = 1 | − 3-s − 5-s + 3·7-s + 9-s − 2·11-s + 2·13-s + 15-s − 17-s + 6·19-s − 3·21-s − 23-s + 25-s − 27-s + 3·29-s − 3·31-s + 2·33-s − 3·35-s + 3·37-s − 2·39-s − 7·41-s − 45-s + 2·49-s + 51-s − 5·53-s + 2·55-s − 6·57-s + 9·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 0.258·15-s − 0.242·17-s + 1.37·19-s − 0.654·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.557·29-s − 0.538·31-s + 0.348·33-s − 0.507·35-s + 0.493·37-s − 0.320·39-s − 1.09·41-s − 0.149·45-s + 2/7·49-s + 0.140·51-s − 0.686·53-s + 0.269·55-s − 0.794·57-s + 1.17·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.547510434\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.547510434\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.600603493441043108112734992629, −8.051678688285955756359172584900, −7.37287096715256720923674298587, −6.57782477277849510090399409127, −5.50261515686804496666550153026, −5.05400746324977715175317429041, −4.18587159900465403118560200563, −3.21608552083442073302077741898, −1.93453494393080793716491508792, −0.822655303884078211967146289953,
0.822655303884078211967146289953, 1.93453494393080793716491508792, 3.21608552083442073302077741898, 4.18587159900465403118560200563, 5.05400746324977715175317429041, 5.50261515686804496666550153026, 6.57782477277849510090399409127, 7.37287096715256720923674298587, 8.051678688285955756359172584900, 8.600603493441043108112734992629