| L(s) = 1 | − 3-s − 5·7-s + 9-s − 5·11-s − 13-s − 3·17-s + 4·19-s + 5·21-s + 5·23-s − 27-s + 4·29-s + 5·33-s + 7·37-s + 39-s + 11·41-s + 12·43-s + 6·47-s + 18·49-s + 3·51-s − 53-s − 4·57-s − 12·59-s + 7·61-s − 5·63-s − 4·67-s − 5·69-s − 7·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.88·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s − 0.727·17-s + 0.917·19-s + 1.09·21-s + 1.04·23-s − 0.192·27-s + 0.742·29-s + 0.870·33-s + 1.15·37-s + 0.160·39-s + 1.71·41-s + 1.82·43-s + 0.875·47-s + 18/7·49-s + 0.420·51-s − 0.137·53-s − 0.529·57-s − 1.56·59-s + 0.896·61-s − 0.629·63-s − 0.488·67-s − 0.601·69-s − 0.830·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9881681906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9881681906\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.06841834251549, −13.67836805489337, −12.99784026868604, −12.82127258692678, −12.49945571520379, −11.76233421783435, −11.15285543547952, −10.63269560403113, −10.26424860829858, −9.713136150650387, −9.191339335277397, −8.862250555530726, −7.837692323645034, −7.363994519322160, −7.060588858224556, −6.228312497618771, −5.892077171048054, −5.435595033819698, −4.567714737985983, −4.221956937238725, −3.172865260553151, −2.823038671928265, −2.365429694178106, −0.9844871708355888, −0.4228559436044133,
0.4228559436044133, 0.9844871708355888, 2.365429694178106, 2.823038671928265, 3.172865260553151, 4.221956937238725, 4.567714737985983, 5.435595033819698, 5.892077171048054, 6.228312497618771, 7.060588858224556, 7.363994519322160, 7.837692323645034, 8.862250555530726, 9.191339335277397, 9.713136150650387, 10.26424860829858, 10.63269560403113, 11.15285543547952, 11.76233421783435, 12.49945571520379, 12.82127258692678, 12.99784026868604, 13.67836805489337, 14.06841834251549
