| L(s) = 1 | + 3-s + 5-s − 2·9-s + 11-s + 4·13-s + 15-s + 6·17-s + 2·19-s + 23-s − 4·25-s − 5·27-s + 2·29-s + 31-s + 33-s − 9·37-s + 4·39-s − 6·41-s + 8·43-s − 2·45-s + 8·47-s + 6·51-s + 10·53-s + 55-s + 2·57-s − 59-s + 2·61-s + 4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 2/3·9-s + 0.301·11-s + 1.10·13-s + 0.258·15-s + 1.45·17-s + 0.458·19-s + 0.208·23-s − 4/5·25-s − 0.962·27-s + 0.371·29-s + 0.179·31-s + 0.174·33-s − 1.47·37-s + 0.640·39-s − 0.937·41-s + 1.21·43-s − 0.298·45-s + 1.16·47-s + 0.840·51-s + 1.37·53-s + 0.134·55-s + 0.264·57-s − 0.130·59-s + 0.256·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.500931279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.500931279\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
| 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
−8.974185337329099643799769666493, −8.407740849273713562840840409918, −7.67917403613721495870291611195, −6.74045693398578522613438993468, −5.76739181125818504000599093049, −5.36066330999113972421117879141, −3.88639452690941288549937743531, −3.31570971349033004173618391007, −2.25052098719008434454447482785, −1.07580389213150519212384849326,
1.07580389213150519212384849326, 2.25052098719008434454447482785, 3.31570971349033004173618391007, 3.88639452690941288549937743531, 5.36066330999113972421117879141, 5.76739181125818504000599093049, 6.74045693398578522613438993468, 7.67917403613721495870291611195, 8.407740849273713562840840409918, 8.974185337329099643799769666493
