| L(s) = 1 | − 2.30·2-s + 3-s + 3.32·4-s − 5-s − 2.30·6-s + 0.182·7-s − 3.04·8-s + 9-s + 2.30·10-s − 5.23·11-s + 3.32·12-s − 2.40·13-s − 0.419·14-s − 15-s + 0.388·16-s − 4.52·17-s − 2.30·18-s − 1.16·19-s − 3.32·20-s + 0.182·21-s + 12.0·22-s + 4.97·23-s − 3.04·24-s + 25-s + 5.54·26-s + 27-s + 0.604·28-s + ⋯ |
| L(s) = 1 | − 1.63·2-s + 0.577·3-s + 1.66·4-s − 0.447·5-s − 0.941·6-s + 0.0688·7-s − 1.07·8-s + 0.333·9-s + 0.729·10-s − 1.57·11-s + 0.958·12-s − 0.667·13-s − 0.112·14-s − 0.258·15-s + 0.0971·16-s − 1.09·17-s − 0.543·18-s − 0.268·19-s − 0.742·20-s + 0.0397·21-s + 2.57·22-s + 1.03·23-s − 0.622·24-s + 0.200·25-s + 1.08·26-s + 0.192·27-s + 0.114·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4985552436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4985552436\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 7 | \( 1 - 0.182T + 7T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 + 2.40T + 13T^{2} \) |
| 17 | \( 1 + 4.52T + 17T^{2} \) |
| 19 | \( 1 + 1.16T + 19T^{2} \) |
| 23 | \( 1 - 4.97T + 23T^{2} \) |
| 29 | \( 1 - 2.17T + 29T^{2} \) |
| 31 | \( 1 - 4.90T + 31T^{2} \) |
| 37 | \( 1 - 3.12T + 37T^{2} \) |
| 41 | \( 1 + 5.11T + 41T^{2} \) |
| 43 | \( 1 + 6.04T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 5.76T + 53T^{2} \) |
| 59 | \( 1 + 2.45T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 1.17T + 67T^{2} \) |
| 71 | \( 1 - 0.741T + 71T^{2} \) |
| 73 | \( 1 - 5.03T + 73T^{2} \) |
| 79 | \( 1 + 1.55T + 79T^{2} \) |
| 83 | \( 1 + 2.80T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−8.267238999874677207757323400214, −7.68426587722454571182336224774, −6.96416175381536209183393559442, −6.45875444616711821192344788323, −5.03012988863862970581679723419, −4.59745815281147516515816540639, −3.18028116266323095051792662836, −2.56052272154771901012590305891, −1.75852866908289136200408695573, −0.44445418357892680180676195830,
0.44445418357892680180676195830, 1.75852866908289136200408695573, 2.56052272154771901012590305891, 3.18028116266323095051792662836, 4.59745815281147516515816540639, 5.03012988863862970581679723419, 6.45875444616711821192344788323, 6.96416175381536209183393559442, 7.68426587722454571182336224774, 8.267238999874677207757323400214
