| L(s) = 1 | + 2.10·3-s − 4.18·5-s + 7-s + 1.42·9-s − 11-s + 13-s − 8.79·15-s − 3.03·17-s + 0.281·19-s + 2.10·21-s − 1.36·23-s + 12.5·25-s − 3.32·27-s + 0.771·29-s − 3.86·31-s − 2.10·33-s − 4.18·35-s + 10.9·37-s + 2.10·39-s + 7.07·41-s + 3.03·43-s − 5.94·45-s + 11.7·47-s + 49-s − 6.39·51-s − 8.75·53-s + 4.18·55-s + ⋯ |
| L(s) = 1 | + 1.21·3-s − 1.87·5-s + 0.377·7-s + 0.473·9-s − 0.301·11-s + 0.277·13-s − 2.27·15-s − 0.737·17-s + 0.0645·19-s + 0.458·21-s − 0.285·23-s + 2.50·25-s − 0.639·27-s + 0.143·29-s − 0.694·31-s − 0.365·33-s − 0.707·35-s + 1.79·37-s + 0.336·39-s + 1.10·41-s + 0.463·43-s − 0.886·45-s + 1.71·47-s + 0.142·49-s − 0.894·51-s − 1.20·53-s + 0.564·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.844760858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.844760858\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 2.10T + 3T^{2} \) |
| 5 | \( 1 + 4.18T + 5T^{2} \) |
| 17 | \( 1 + 3.03T + 17T^{2} \) |
| 19 | \( 1 - 0.281T + 19T^{2} \) |
| 23 | \( 1 + 1.36T + 23T^{2} \) |
| 29 | \( 1 - 0.771T + 29T^{2} \) |
| 31 | \( 1 + 3.86T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 7.07T + 41T^{2} \) |
| 43 | \( 1 - 3.03T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 8.75T + 53T^{2} \) |
| 59 | \( 1 - 2.35T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 9.82T + 67T^{2} \) |
| 71 | \( 1 + 3.91T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 9.27T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 + 6.61T + 97T^{2} \) |
| 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
−8.369877164147699881303258489532, −7.72915466532242203110767470319, −7.47830098279510131433419293786, −6.45878952578568907845745169448, −5.28921098930746516951457488543, −4.18379018284109465959932519435, −3.98491221350448650818305992643, −3.00082285369161385549359382839, −2.27149474331131056345072238116, −0.71868963667970795682473811754,
0.71868963667970795682473811754, 2.27149474331131056345072238116, 3.00082285369161385549359382839, 3.98491221350448650818305992643, 4.18379018284109465959932519435, 5.28921098930746516951457488543, 6.45878952578568907845745169448, 7.47830098279510131433419293786, 7.72915466532242203110767470319, 8.369877164147699881303258489532
