| L(s) = 1 | + 0.728·2-s + 2.62·3-s − 1.46·4-s − 5-s + 1.91·6-s + 2.55·7-s − 2.52·8-s + 3.91·9-s − 0.728·10-s + 2.46·11-s − 3.86·12-s + 1.55·13-s + 1.86·14-s − 2.62·15-s + 1.09·16-s − 6.83·17-s + 2.85·18-s − 7.66·19-s + 1.46·20-s + 6.70·21-s + 1.79·22-s − 7.50·23-s − 6.64·24-s + 25-s + 1.13·26-s + 2.39·27-s − 3.74·28-s + ⋯ |
| L(s) = 1 | + 0.515·2-s + 1.51·3-s − 0.734·4-s − 0.447·5-s + 0.782·6-s + 0.964·7-s − 0.893·8-s + 1.30·9-s − 0.230·10-s + 0.744·11-s − 1.11·12-s + 0.432·13-s + 0.497·14-s − 0.678·15-s + 0.273·16-s − 1.65·17-s + 0.671·18-s − 1.75·19-s + 0.328·20-s + 1.46·21-s + 0.383·22-s − 1.56·23-s − 1.35·24-s + 0.200·25-s + 0.222·26-s + 0.460·27-s − 0.708·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.898215108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.898215108\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 - 0.728T + 2T^{2} \) |
| 3 | \( 1 - 2.62T + 3T^{2} \) |
| 7 | \( 1 - 2.55T + 7T^{2} \) |
| 11 | \( 1 - 2.46T + 11T^{2} \) |
| 13 | \( 1 - 1.55T + 13T^{2} \) |
| 17 | \( 1 + 6.83T + 17T^{2} \) |
| 19 | \( 1 + 7.66T + 19T^{2} \) |
| 23 | \( 1 + 7.50T + 23T^{2} \) |
| 29 | \( 1 - 3.25T + 29T^{2} \) |
| 31 | \( 1 - 0.658T + 31T^{2} \) |
| 41 | \( 1 - 2.46T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 - 3.11T + 47T^{2} \) |
| 53 | \( 1 - 8.64T + 53T^{2} \) |
| 59 | \( 1 + 6.23T + 59T^{2} \) |
| 61 | \( 1 - 3.27T + 61T^{2} \) |
| 67 | \( 1 - 1.47T + 67T^{2} \) |
| 71 | \( 1 + 8.06T + 71T^{2} \) |
| 73 | \( 1 + 4.96T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 1.14T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−12.93797098730614069988442852302, −11.85537848149979886481712301094, −10.61712625414538025213881089270, −9.128639057507731061676352667476, −8.637437220944088834840893018036, −7.891017107362987824800055775578, −6.30728224565534320902229750063, −4.38470732634730556296851103179, −3.98850568726270041730242913214, −2.26433154409388450377348441114,
2.26433154409388450377348441114, 3.98850568726270041730242913214, 4.38470732634730556296851103179, 6.30728224565534320902229750063, 7.891017107362987824800055775578, 8.637437220944088834840893018036, 9.128639057507731061676352667476, 10.61712625414538025213881089270, 11.85537848149979886481712301094, 12.93797098730614069988442852302
