| L(s) = 1 | + 2.32·3-s + 5-s − 4.78·7-s + 2.40·9-s − 0.601·13-s + 2.32·15-s − 2.64·17-s − 2.60·19-s − 11.1·21-s − 5.17·23-s + 25-s − 1.38·27-s + 5.44·29-s + 10.4·31-s − 4.78·35-s + 9.39·37-s − 1.39·39-s − 4.54·41-s + 3.48·43-s + 2.40·45-s − 5.07·47-s + 15.9·49-s − 6.15·51-s + 3.76·53-s − 6.04·57-s − 11.7·59-s + 12.6·61-s + ⋯ |
| L(s) = 1 | + 1.34·3-s + 0.447·5-s − 1.80·7-s + 0.801·9-s − 0.166·13-s + 0.600·15-s − 0.642·17-s − 0.596·19-s − 2.42·21-s − 1.07·23-s + 0.200·25-s − 0.266·27-s + 1.01·29-s + 1.87·31-s − 0.809·35-s + 1.54·37-s − 0.223·39-s − 0.710·41-s + 0.531·43-s + 0.358·45-s − 0.740·47-s + 2.27·49-s − 0.862·51-s + 0.517·53-s − 0.801·57-s − 1.53·59-s + 1.61·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.554312795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.554312795\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2.32T + 3T^{2} \) |
| 7 | \( 1 + 4.78T + 7T^{2} \) |
| 13 | \( 1 + 0.601T + 13T^{2} \) |
| 17 | \( 1 + 2.64T + 17T^{2} \) |
| 19 | \( 1 + 2.60T + 19T^{2} \) |
| 23 | \( 1 + 5.17T + 23T^{2} \) |
| 29 | \( 1 - 5.44T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 9.39T + 37T^{2} \) |
| 41 | \( 1 + 4.54T + 41T^{2} \) |
| 43 | \( 1 - 3.48T + 43T^{2} \) |
| 47 | \( 1 + 5.07T + 47T^{2} \) |
| 53 | \( 1 - 3.76T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 - 15.0T + 67T^{2} \) |
| 71 | \( 1 + 0.346T + 71T^{2} \) |
| 73 | \( 1 - 3.68T + 73T^{2} \) |
| 79 | \( 1 - 8.05T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 0.462T + 89T^{2} \) |
| 97 | \( 1 - 7.97T + 97T^{2} \) |
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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
−7.917022150258502212221185025502, −6.84084249636301445704364349307, −6.44975802905762280024024851340, −5.91506896717222042583471931035, −4.71762116732611156016846638592, −3.96747901105752402282481375036, −3.28832547874418265050022479538, −2.58238519748532288591885803448, −2.18187443758053782390324875574, −0.67308080085975658562349568071,
0.67308080085975658562349568071, 2.18187443758053782390324875574, 2.58238519748532288591885803448, 3.28832547874418265050022479538, 3.96747901105752402282481375036, 4.71762116732611156016846638592, 5.91506896717222042583471931035, 6.44975802905762280024024851340, 6.84084249636301445704364349307, 7.917022150258502212221185025502
