L(s) = 1 | + 2.80·3-s − 3.33·5-s + 2.85·7-s + 4.87·9-s + 6.16·11-s − 3.18·13-s − 9.34·15-s − 3.44·17-s − 6.59·19-s + 8.01·21-s − 9.22·23-s + 6.09·25-s + 5.27·27-s + 1.40·29-s − 0.702·31-s + 17.2·33-s − 9.51·35-s − 9.54·37-s − 8.93·39-s − 2.82·41-s − 8.28·43-s − 16.2·45-s − 10.2·47-s + 1.16·49-s − 9.67·51-s − 5.03·53-s − 20.5·55-s + ⋯ |
L(s) = 1 | + 1.62·3-s − 1.48·5-s + 1.07·7-s + 1.62·9-s + 1.85·11-s − 0.882·13-s − 2.41·15-s − 0.836·17-s − 1.51·19-s + 1.75·21-s − 1.92·23-s + 1.21·25-s + 1.01·27-s + 0.260·29-s − 0.126·31-s + 3.01·33-s − 1.60·35-s − 1.56·37-s − 1.43·39-s − 0.441·41-s − 1.26·43-s − 2.42·45-s − 1.49·47-s + 0.166·49-s − 1.35·51-s − 0.691·53-s − 2.76·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
good | 3 | \( 1 - 2.80T + 3T^{2} \) |
| 5 | \( 1 + 3.33T + 5T^{2} \) |
| 7 | \( 1 - 2.85T + 7T^{2} \) |
| 11 | \( 1 - 6.16T + 11T^{2} \) |
| 13 | \( 1 + 3.18T + 13T^{2} \) |
| 17 | \( 1 + 3.44T + 17T^{2} \) |
| 19 | \( 1 + 6.59T + 19T^{2} \) |
| 23 | \( 1 + 9.22T + 23T^{2} \) |
| 29 | \( 1 - 1.40T + 29T^{2} \) |
| 31 | \( 1 + 0.702T + 31T^{2} \) |
| 37 | \( 1 + 9.54T + 37T^{2} \) |
| 41 | \( 1 + 2.82T + 41T^{2} \) |
| 43 | \( 1 + 8.28T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 5.03T + 53T^{2} \) |
| 59 | \( 1 - 3.57T + 59T^{2} \) |
| 61 | \( 1 - 2.46T + 61T^{2} \) |
| 67 | \( 1 + 0.271T + 67T^{2} \) |
| 71 | \( 1 - 6.18T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 - 7.67T + 79T^{2} \) |
| 83 | \( 1 + 1.85T + 83T^{2} \) |
| 89 | \( 1 + 1.46T + 89T^{2} \) |
| 97 | \( 1 + 0.763T + 97T^{2} \) |
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
−7.83905960758360221622316743428, −6.82709151253750848973903082323, −6.62146158914302325158313126897, −4.99373789614642235444073129468, −4.23804951798014758262408057551, −3.97685722065148951268309920161, −3.30457637641070027422658644464, −2.07017138769883736619352943805, −1.70624854843769854778435334651, 0,
1.70624854843769854778435334651, 2.07017138769883736619352943805, 3.30457637641070027422658644464, 3.97685722065148951268309920161, 4.23804951798014758262408057551, 4.99373789614642235444073129468, 6.62146158914302325158313126897, 6.82709151253750848973903082323, 7.83905960758360221622316743428