| L(s) = 1 | − 3-s − 1.52·5-s + 7-s + 9-s − 1.86·11-s + 0.604·13-s + 1.52·15-s − 2.92·17-s + 1.86·19-s − 21-s + 23-s − 2.66·25-s − 27-s + 7.19·29-s + 5.86·31-s + 1.86·33-s − 1.52·35-s − 3.19·37-s − 0.604·39-s − 6.65·41-s + 5.66·43-s − 1.52·45-s − 10.7·47-s + 49-s + 2.92·51-s − 1.52·53-s + 2.85·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.683·5-s + 0.377·7-s + 0.333·9-s − 0.562·11-s + 0.167·13-s + 0.394·15-s − 0.709·17-s + 0.428·19-s − 0.218·21-s + 0.208·23-s − 0.532·25-s − 0.192·27-s + 1.33·29-s + 1.05·31-s + 0.324·33-s − 0.258·35-s − 0.524·37-s − 0.0968·39-s − 1.03·41-s + 0.863·43-s − 0.227·45-s − 1.57·47-s + 0.142·49-s + 0.409·51-s − 0.210·53-s + 0.384·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.108144082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.108144082\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 1.52T + 5T^{2} \) |
| 11 | \( 1 + 1.86T + 11T^{2} \) |
| 13 | \( 1 - 0.604T + 13T^{2} \) |
| 17 | \( 1 + 2.92T + 17T^{2} \) |
| 19 | \( 1 - 1.86T + 19T^{2} \) |
| 29 | \( 1 - 7.19T + 29T^{2} \) |
| 31 | \( 1 - 5.86T + 31T^{2} \) |
| 37 | \( 1 + 3.19T + 37T^{2} \) |
| 41 | \( 1 + 6.65T + 41T^{2} \) |
| 43 | \( 1 - 5.66T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 1.52T + 53T^{2} \) |
| 59 | \( 1 - 6.31T + 59T^{2} \) |
| 61 | \( 1 + 3.66T + 61T^{2} \) |
| 67 | \( 1 + 5.37T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 + 9.19T + 79T^{2} \) |
| 83 | \( 1 - 0.924T + 83T^{2} \) |
| 89 | \( 1 - 7.39T + 89T^{2} \) |
| 97 | \( 1 - 5.98T + 97T^{2} \) |
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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
−7.906027758133638706035286735519, −7.14987501411402489783279258206, −6.50956496658022966666959761432, −5.78720895441251495112741824511, −4.87293144868399731247894068704, −4.54820595190026370507019851860, −3.58005816341294718145395960014, −2.74258983520197288409493465291, −1.66021402336329761698448652519, −0.54634687373185361043925360645,
0.54634687373185361043925360645, 1.66021402336329761698448652519, 2.74258983520197288409493465291, 3.58005816341294718145395960014, 4.54820595190026370507019851860, 4.87293144868399731247894068704, 5.78720895441251495112741824511, 6.50956496658022966666959761432, 7.14987501411402489783279258206, 7.906027758133638706035286735519
