| L(s) = 1 | − 2-s + 4-s + 4.04·5-s − 0.153·7-s − 8-s − 4.04·10-s − 0.220·11-s + 0.594·13-s + 0.153·14-s + 16-s − 5.56·17-s + 3.82·19-s + 4.04·20-s + 0.220·22-s + 5.85·23-s + 11.3·25-s − 0.594·26-s − 0.153·28-s − 0.747·29-s + 5.08·31-s − 32-s + 5.56·34-s − 0.621·35-s + 3.74·37-s − 3.82·38-s − 4.04·40-s − 3.97·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.81·5-s − 0.0580·7-s − 0.353·8-s − 1.27·10-s − 0.0666·11-s + 0.164·13-s + 0.0410·14-s + 0.250·16-s − 1.34·17-s + 0.876·19-s + 0.905·20-s + 0.0470·22-s + 1.22·23-s + 2.27·25-s − 0.116·26-s − 0.0290·28-s − 0.138·29-s + 0.913·31-s − 0.176·32-s + 0.954·34-s − 0.105·35-s + 0.616·37-s − 0.619·38-s − 0.639·40-s − 0.620·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.746255115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.746255115\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 4.04T + 5T^{2} \) |
| 7 | \( 1 + 0.153T + 7T^{2} \) |
| 11 | \( 1 + 0.220T + 11T^{2} \) |
| 13 | \( 1 - 0.594T + 13T^{2} \) |
| 17 | \( 1 + 5.56T + 17T^{2} \) |
| 19 | \( 1 - 3.82T + 19T^{2} \) |
| 23 | \( 1 - 5.85T + 23T^{2} \) |
| 29 | \( 1 + 0.747T + 29T^{2} \) |
| 31 | \( 1 - 5.08T + 31T^{2} \) |
| 37 | \( 1 - 3.74T + 37T^{2} \) |
| 41 | \( 1 + 3.97T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 0.475T + 47T^{2} \) |
| 53 | \( 1 + 3.29T + 53T^{2} \) |
| 59 | \( 1 + 7.27T + 59T^{2} \) |
| 61 | \( 1 - 3.36T + 61T^{2} \) |
| 67 | \( 1 - 7.45T + 67T^{2} \) |
| 71 | \( 1 + 2.85T + 71T^{2} \) |
| 73 | \( 1 + 1.27T + 73T^{2} \) |
| 79 | \( 1 + 3.30T + 79T^{2} \) |
| 83 | \( 1 - 8.27T + 83T^{2} \) |
| 89 | \( 1 - 4.95T + 89T^{2} \) |
| 97 | \( 1 - 5.65T + 97T^{2} \) |
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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
−9.266523141074700090788357926086, −9.165277708868846630065471074489, −8.052267975981998889488798071422, −6.92967297460183396671823764629, −6.38154068837441365016760183393, −5.55830601772581475853888739596, −4.67839283508974520653305776341, −3.01168271511900394553628151964, −2.19671977185143841113890969216, −1.12840431730195458026621346978,
1.12840431730195458026621346978, 2.19671977185143841113890969216, 3.01168271511900394553628151964, 4.67839283508974520653305776341, 5.55830601772581475853888739596, 6.38154068837441365016760183393, 6.92967297460183396671823764629, 8.052267975981998889488798071422, 9.165277708868846630065471074489, 9.266523141074700090788357926086
