| L(s) = 1 | − 3-s + 7-s + 9-s + 11-s + 13-s − 17-s + 19-s − 21-s + 23-s − 27-s − 29-s − 31-s − 33-s + 37-s − 39-s + 41-s − 43-s + 47-s + 49-s + 51-s + 53-s − 57-s + 59-s − 61-s + 63-s − 67-s − 69-s + ⋯ |
| L(s) = 1 | − 3-s + 7-s + 9-s + 11-s + 13-s − 17-s + 19-s − 21-s + 23-s − 27-s − 29-s − 31-s − 33-s + 37-s − 39-s + 41-s − 43-s + 47-s + 49-s + 51-s + 53-s − 57-s + 59-s − 61-s + 63-s − 67-s − 69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.273701528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.273701528\) |
| \(L(1)\) |
\(\approx\) |
\(0.9934588265\) |
| \(L(1)\) |
\(\approx\) |
\(0.9934588265\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−34.89620878693812615394724706126, −33.47177184188358054648343802471, −32.95373110336489202294647741903, −31.000878238848054813734736914031, −30.08283072310760662654485254361, −28.75620407320413245889633013070, −27.74001549128314964333399015308, −26.79999431487267833684377135315, −24.91164474281862795156104112947, −23.94125759717354704293742119293, −22.71111814469318191049357358124, −21.62820070972895871809325135529, −20.30870606713841120462503925791, −18.50131456917208069943603260787, −17.56198392586458758656427315753, −16.38749542986661449060865447244, −14.9334623468118711546352731953, −13.31089863478015673369236422861, −11.66929573512790455200503900855, −10.93602082108279406030214194985, −9.06078728593525702719913749440, −7.19780040148562061479699240603, −5.69697314536849519104203193886, −4.211079232419059888877011668122, −1.32058972400961870327719439444,
1.32058972400961870327719439444, 4.211079232419059888877011668122, 5.69697314536849519104203193886, 7.19780040148562061479699240603, 9.06078728593525702719913749440, 10.93602082108279406030214194985, 11.66929573512790455200503900855, 13.31089863478015673369236422861, 14.9334623468118711546352731953, 16.38749542986661449060865447244, 17.56198392586458758656427315753, 18.50131456917208069943603260787, 20.30870606713841120462503925791, 21.62820070972895871809325135529, 22.71111814469318191049357358124, 23.94125759717354704293742119293, 24.91164474281862795156104112947, 26.79999431487267833684377135315, 27.74001549128314964333399015308, 28.75620407320413245889633013070, 30.08283072310760662654485254361, 31.000878238848054813734736914031, 32.95373110336489202294647741903, 33.47177184188358054648343802471, 34.89620878693812615394724706126
