| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 11-s − 14-s + 16-s + 17-s − 19-s − 22-s + 23-s + 28-s − 29-s − 31-s − 32-s − 34-s + 37-s + 38-s + 41-s − 43-s + 44-s − 46-s − 47-s + 49-s + 53-s − 56-s + 58-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 11-s − 14-s + 16-s + 17-s − 19-s − 22-s + 23-s + 28-s − 29-s − 31-s − 32-s − 34-s + 37-s + 38-s + 41-s − 43-s + 44-s − 46-s − 47-s + 49-s + 53-s − 56-s + 58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.447248236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.447248236\) |
| \(L(1)\) |
\(\approx\) |
\(0.8998964909\) |
| \(L(1)\) |
\(\approx\) |
\(0.8998964909\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 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
−27.05804054254587441811611087090, −25.813020865483933427420755921324, −25.02973657432677463827443108795, −24.20749242549201843740509963031, −23.1838947791430319130370827474, −21.66470041040740636250407992136, −20.93119317851929787493244016784, −19.94639552977000728799853054038, −18.999859233412595536601066345313, −18.11432778231971052887963989480, −17.08580806852820881020623566344, −16.5456781111182323124203970040, −14.973092766118638530246332348920, −14.538848271452150676678298968522, −12.77719430144747225012896796258, −11.59185417688313152061044940430, −10.938158254065847143410105978751, −9.67389896739758428466134234082, −8.70801850268493958294692686673, −7.7514641294761048287883396593, −6.68473852171955440679235617240, −5.38709887279166918439290362722, −3.754981130364287010276866900539, −2.11047334670063318489282953274, −0.980506283014651926272391264938,
0.980506283014651926272391264938, 2.11047334670063318489282953274, 3.754981130364287010276866900539, 5.38709887279166918439290362722, 6.68473852171955440679235617240, 7.7514641294761048287883396593, 8.70801850268493958294692686673, 9.67389896739758428466134234082, 10.938158254065847143410105978751, 11.59185417688313152061044940430, 12.77719430144747225012896796258, 14.538848271452150676678298968522, 14.973092766118638530246332348920, 16.5456781111182323124203970040, 17.08580806852820881020623566344, 18.11432778231971052887963989480, 18.999859233412595536601066345313, 19.94639552977000728799853054038, 20.93119317851929787493244016784, 21.66470041040740636250407992136, 23.1838947791430319130370827474, 24.20749242549201843740509963031, 25.02973657432677463827443108795, 25.813020865483933427420755921324, 27.05804054254587441811611087090
