| L(s) = 1 | + 1.56·3-s + 5-s + 3.56·7-s − 0.561·9-s − 11-s + 3.12·13-s + 1.56·15-s + 5.56·17-s − 2.43·19-s + 5.56·21-s + 7.12·23-s + 25-s − 5.56·27-s + 0.438·29-s + 8.68·31-s − 1.56·33-s + 3.56·35-s − 9.80·37-s + 4.87·39-s − 10·41-s − 5.12·43-s − 0.561·45-s − 7.12·47-s + 5.68·49-s + 8.68·51-s + 4.43·53-s − 55-s + ⋯ |
| L(s) = 1 | + 0.901·3-s + 0.447·5-s + 1.34·7-s − 0.187·9-s − 0.301·11-s + 0.866·13-s + 0.403·15-s + 1.34·17-s − 0.559·19-s + 1.21·21-s + 1.48·23-s + 0.200·25-s − 1.07·27-s + 0.0814·29-s + 1.55·31-s − 0.271·33-s + 0.602·35-s − 1.61·37-s + 0.780·39-s − 1.56·41-s − 0.781·43-s − 0.0837·45-s − 1.03·47-s + 0.812·49-s + 1.21·51-s + 0.609·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.514672933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.514672933\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 7 | \( 1 - 3.56T + 7T^{2} \) |
| 13 | \( 1 - 3.12T + 13T^{2} \) |
| 17 | \( 1 - 5.56T + 17T^{2} \) |
| 19 | \( 1 + 2.43T + 19T^{2} \) |
| 23 | \( 1 - 7.12T + 23T^{2} \) |
| 29 | \( 1 - 0.438T + 29T^{2} \) |
| 31 | \( 1 - 8.68T + 31T^{2} \) |
| 37 | \( 1 + 9.80T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 5.12T + 43T^{2} \) |
| 47 | \( 1 + 7.12T + 47T^{2} \) |
| 53 | \( 1 - 4.43T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 3.56T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 2.43T + 71T^{2} \) |
| 73 | \( 1 - 4.87T + 73T^{2} \) |
| 79 | \( 1 - 0.876T + 79T^{2} \) |
| 83 | \( 1 + 10T + 83T^{2} \) |
| 89 | \( 1 - 9.80T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.355162525083403676785383397804, −8.246827489604441903006458722531, −7.23585986510385669341080601304, −6.38446455160626141403482528478, −5.31162932849553428975449686695, −4.96553277235539371369580082976, −3.70017216051255956833919344856, −3.01720714862118923861259366411, −2.01232064853539611645643860034, −1.17526251650271530336063512894,
1.17526251650271530336063512894, 2.01232064853539611645643860034, 3.01720714862118923861259366411, 3.70017216051255956833919344856, 4.96553277235539371369580082976, 5.31162932849553428975449686695, 6.38446455160626141403482528478, 7.23585986510385669341080601304, 8.246827489604441903006458722531, 8.355162525083403676785383397804
