| L(s) = 1 | + 3.16·3-s + 2·5-s − 3.16·7-s + 7.00·9-s + 3.16·11-s + 4·13-s + 6.32·15-s − 17-s − 6.32·19-s − 10.0·21-s + 3.16·23-s − 25-s + 12.6·27-s − 10·29-s − 3.16·31-s + 10.0·33-s − 6.32·35-s + 2·37-s + 12.6·39-s + 10·41-s − 6.32·43-s + 14.0·45-s + 3.00·49-s − 3.16·51-s + 6·53-s + 6.32·55-s − 20.0·57-s + ⋯ |
| L(s) = 1 | + 1.82·3-s + 0.894·5-s − 1.19·7-s + 2.33·9-s + 0.953·11-s + 1.10·13-s + 1.63·15-s − 0.242·17-s − 1.45·19-s − 2.18·21-s + 0.659·23-s − 0.200·25-s + 2.43·27-s − 1.85·29-s − 0.567·31-s + 1.74·33-s − 1.06·35-s + 0.328·37-s + 2.02·39-s + 1.56·41-s − 0.964·43-s + 2.08·45-s + 0.428·49-s − 0.442·51-s + 0.824·53-s + 0.852·55-s − 2.64·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.331510512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.331510512\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 3.16T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + 3.16T + 7T^{2} \) |
| 11 | \( 1 - 3.16T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 19 | \( 1 + 6.32T + 19T^{2} \) |
| 23 | \( 1 - 3.16T + 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 + 3.16T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 6.32T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 6.32T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 - 3.16T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 9.48T + 79T^{2} \) |
| 83 | \( 1 + 6.32T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−9.582131902200732341534881066127, −9.018538973351128834491889676080, −8.611015310627638039109274462834, −7.40276959290589380910453278935, −6.59683303760921539412808925275, −5.87114349917837056344535544147, −4.09744184484786018126878034880, −3.59660581176879197521221719493, −2.51392513984655019136943107621, −1.60880597920637733702523750134,
1.60880597920637733702523750134, 2.51392513984655019136943107621, 3.59660581176879197521221719493, 4.09744184484786018126878034880, 5.87114349917837056344535544147, 6.59683303760921539412808925275, 7.40276959290589380910453278935, 8.611015310627638039109274462834, 9.018538973351128834491889676080, 9.582131902200732341534881066127
