| L(s) = 1 | − 0.198·3-s + 2.80·5-s − 2.96·9-s + 0.643·11-s + 5.49·13-s − 0.554·15-s − 2.58·17-s − 6.60·19-s − 2.85·23-s + 2.85·25-s + 1.18·27-s − 9.74·29-s + 5.14·31-s − 0.127·33-s − 2.30·37-s − 1.08·39-s − 4.82·41-s − 6.32·43-s − 8.29·45-s − 12.2·47-s + 0.511·51-s − 6.32·53-s + 1.80·55-s + 1.30·57-s + 5.96·59-s − 3.56·61-s + 15.3·65-s + ⋯ |
| L(s) = 1 | − 0.114·3-s + 1.25·5-s − 0.986·9-s + 0.193·11-s + 1.52·13-s − 0.143·15-s − 0.626·17-s − 1.51·19-s − 0.594·23-s + 0.570·25-s + 0.227·27-s − 1.80·29-s + 0.924·31-s − 0.0221·33-s − 0.379·37-s − 0.174·39-s − 0.753·41-s − 0.964·43-s − 1.23·45-s − 1.79·47-s + 0.0716·51-s − 0.868·53-s + 0.242·55-s + 0.173·57-s + 0.776·59-s − 0.456·61-s + 1.90·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.198T + 3T^{2} \) |
| 5 | \( 1 - 2.80T + 5T^{2} \) |
| 11 | \( 1 - 0.643T + 11T^{2} \) |
| 13 | \( 1 - 5.49T + 13T^{2} \) |
| 17 | \( 1 + 2.58T + 17T^{2} \) |
| 19 | \( 1 + 6.60T + 19T^{2} \) |
| 23 | \( 1 + 2.85T + 23T^{2} \) |
| 29 | \( 1 + 9.74T + 29T^{2} \) |
| 31 | \( 1 - 5.14T + 31T^{2} \) |
| 37 | \( 1 + 2.30T + 37T^{2} \) |
| 41 | \( 1 + 4.82T + 41T^{2} \) |
| 43 | \( 1 + 6.32T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 + 6.32T + 53T^{2} \) |
| 59 | \( 1 - 5.96T + 59T^{2} \) |
| 61 | \( 1 + 3.56T + 61T^{2} \) |
| 67 | \( 1 - 9.18T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 9.53T + 73T^{2} \) |
| 79 | \( 1 - 1.47T + 79T^{2} \) |
| 83 | \( 1 - 3.80T + 83T^{2} \) |
| 89 | \( 1 + 4.95T + 89T^{2} \) |
| 97 | \( 1 + 7.85T + 97T^{2} \) |
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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
−8.120831101439254474020280155103, −6.67975116652366110850960470607, −6.36014393191180166039875983054, −5.78280147626112034166937686662, −5.07527386779439562529306915740, −4.04449015127557733996973063761, −3.24735232324960115503831794783, −2.15982874298337689632294714916, −1.59190440418170957430267468763, 0,
1.59190440418170957430267468763, 2.15982874298337689632294714916, 3.24735232324960115503831794783, 4.04449015127557733996973063761, 5.07527386779439562529306915740, 5.78280147626112034166937686662, 6.36014393191180166039875983054, 6.67975116652366110850960470607, 8.120831101439254474020280155103
