| L(s) = 1 | − 2.96·2-s + 0.778·4-s − 20.0·5-s − 4.37·7-s + 21.3·8-s + 59.4·10-s − 18.0·11-s + 82.1·13-s + 12.9·14-s − 69.6·16-s + 100.·17-s + 43.8·19-s − 15.6·20-s + 53.4·22-s − 19.2·23-s + 277.·25-s − 243.·26-s − 3.40·28-s + 46.3·29-s + 307.·31-s + 35.1·32-s − 296.·34-s + 87.8·35-s − 176.·37-s − 129.·38-s − 429.·40-s + 15.7·41-s + ⋯ |
| L(s) = 1 | − 1.04·2-s + 0.0973·4-s − 1.79·5-s − 0.236·7-s + 0.945·8-s + 1.88·10-s − 0.494·11-s + 1.75·13-s + 0.247·14-s − 1.08·16-s + 1.42·17-s + 0.529·19-s − 0.174·20-s + 0.518·22-s − 0.174·23-s + 2.22·25-s − 1.83·26-s − 0.0229·28-s + 0.296·29-s + 1.78·31-s + 0.193·32-s − 1.49·34-s + 0.424·35-s − 0.785·37-s − 0.554·38-s − 1.69·40-s + 0.0598·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8378651899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8378651899\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 + 2.96T + 8T^{2} \) |
| 5 | \( 1 + 20.0T + 125T^{2} \) |
| 7 | \( 1 + 4.37T + 343T^{2} \) |
| 11 | \( 1 + 18.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 82.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 100.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 43.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 19.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 46.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 307.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 176.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 15.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 483.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 260.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 217.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 176.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 324.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.07e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 236.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 343.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 326.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 245.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.45e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 762.T + 9.12e5T^{2} \) |
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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
−8.411437780969533279172966827958, −8.166705423043797704431328024542, −7.55334305418772014202623672563, −6.72878069561675333893475978586, −5.58250258513208405178449341981, −4.48850819003220189933635532849, −3.77592883508306492320268988761, −3.00515175283656938144887511954, −1.18330854041584610259797158556, −0.60366262437006772728958471626,
0.60366262437006772728958471626, 1.18330854041584610259797158556, 3.00515175283656938144887511954, 3.77592883508306492320268988761, 4.48850819003220189933635532849, 5.58250258513208405178449341981, 6.72878069561675333893475978586, 7.55334305418772014202623672563, 8.166705423043797704431328024542, 8.411437780969533279172966827958
