| L(s) = 1 | + 2-s + 4-s − 1.07·7-s + 8-s − 3.41·11-s − 0.921·13-s − 1.07·14-s + 16-s − 0.340·17-s − 19-s − 3.41·22-s + 0.921·23-s − 0.921·26-s − 1.07·28-s + 1.41·29-s + 7.26·31-s + 32-s − 0.340·34-s + 5.60·37-s − 38-s + 1.07·41-s + 0.738·43-s − 3.41·44-s + 0.921·46-s + 7.75·47-s − 5.83·49-s − 0.921·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.407·7-s + 0.353·8-s − 1.03·11-s − 0.255·13-s − 0.288·14-s + 0.250·16-s − 0.0825·17-s − 0.229·19-s − 0.728·22-s + 0.192·23-s − 0.180·26-s − 0.203·28-s + 0.263·29-s + 1.30·31-s + 0.176·32-s − 0.0583·34-s + 0.920·37-s − 0.162·38-s + 0.168·41-s + 0.112·43-s − 0.515·44-s + 0.135·46-s + 1.13·47-s − 0.833·49-s − 0.127·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.685883002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.685883002\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 1.07T + 7T^{2} \) |
| 11 | \( 1 + 3.41T + 11T^{2} \) |
| 13 | \( 1 + 0.921T + 13T^{2} \) |
| 17 | \( 1 + 0.340T + 17T^{2} \) |
| 23 | \( 1 - 0.921T + 23T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 - 7.26T + 31T^{2} \) |
| 37 | \( 1 - 5.60T + 37T^{2} \) |
| 41 | \( 1 - 1.07T + 41T^{2} \) |
| 43 | \( 1 - 0.738T + 43T^{2} \) |
| 47 | \( 1 - 7.75T + 47T^{2} \) |
| 53 | \( 1 + 2.68T + 53T^{2} \) |
| 59 | \( 1 + 8.34T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 4.68T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 + 6.83T + 73T^{2} \) |
| 79 | \( 1 - 4.73T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 9.75T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−7.65876161192441463500016030727, −7.06465137349282106941391203042, −6.20278028738625495626483851238, −5.81227021516801464019663748728, −4.79931281760801231230335785800, −4.48928668182503240168271057512, −3.37809071150711237115101466420, −2.79142223245457426799506676119, −2.03733770360757473493935637481, −0.69543337658002509181844921460,
0.69543337658002509181844921460, 2.03733770360757473493935637481, 2.79142223245457426799506676119, 3.37809071150711237115101466420, 4.48928668182503240168271057512, 4.79931281760801231230335785800, 5.81227021516801464019663748728, 6.20278028738625495626483851238, 7.06465137349282106941391203042, 7.65876161192441463500016030727
