| L(s) = 1 | − 3-s − 7-s − 2·9-s − 3·11-s + 3·13-s − 3·17-s − 2·19-s + 21-s + 23-s + 5·27-s − 3·29-s + 4·31-s + 3·33-s + 8·37-s − 3·39-s − 10·41-s − 4·43-s + 13·47-s + 49-s + 3·51-s + 2·57-s − 8·59-s + 10·61-s + 2·63-s + 6·67-s − 69-s − 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s − 0.904·11-s + 0.832·13-s − 0.727·17-s − 0.458·19-s + 0.218·21-s + 0.208·23-s + 0.962·27-s − 0.557·29-s + 0.718·31-s + 0.522·33-s + 1.31·37-s − 0.480·39-s − 1.56·41-s − 0.609·43-s + 1.89·47-s + 1/7·49-s + 0.420·51-s + 0.264·57-s − 1.04·59-s + 1.28·61-s + 0.251·63-s + 0.733·67-s − 0.120·69-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8128702539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8128702539\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−12.85306316183158, −12.36954748439057, −11.82016679490037, −11.40773611347919, −10.96044697225340, −10.67333849050320, −10.10783581083166, −9.703064777952804, −8.963747051822848, −8.647354778605560, −8.239748459599164, −7.692349537587410, −7.056308858404979, −6.557756539695975, −6.191599306422432, −5.627994002354908, −5.312903993414375, −4.650644023163458, −4.151890325900889, −3.554656520763113, −2.848802780649701, −2.545217070973881, −1.788826967984528, −0.9671137315540613, −0.2934469320971064,
0.2934469320971064, 0.9671137315540613, 1.788826967984528, 2.545217070973881, 2.848802780649701, 3.554656520763113, 4.151890325900889, 4.650644023163458, 5.312903993414375, 5.627994002354908, 6.191599306422432, 6.557756539695975, 7.056308858404979, 7.692349537587410, 8.239748459599164, 8.647354778605560, 8.963747051822848, 9.703064777952804, 10.10783581083166, 10.67333849050320, 10.96044697225340, 11.40773611347919, 11.82016679490037, 12.36954748439057, 12.85306316183158
