L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s + 3·11-s − 12-s + 13-s − 14-s + 16-s + 3·17-s − 18-s − 7·19-s − 21-s − 3·22-s − 3·23-s + 24-s − 26-s − 27-s + 28-s + 2·31-s − 32-s − 3·33-s − 3·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.904·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 1.60·19-s − 0.218·21-s − 0.639·22-s − 0.625·23-s + 0.204·24-s − 0.196·26-s − 0.192·27-s + 0.188·28-s + 0.359·31-s − 0.176·32-s − 0.522·33-s − 0.514·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.001513662168442863978309623759, −7.01437951887825475719301902451, −6.39769483319444291966101608301, −5.90767299024270226404425939091, −4.88110072193851369524093531967, −4.14333102039663126277605679814, −3.24406126368525216574706922173, −1.98471408431458957288316935137, −1.28716270479178369718416467812, 0,
1.28716270479178369718416467812, 1.98471408431458957288316935137, 3.24406126368525216574706922173, 4.14333102039663126277605679814, 4.88110072193851369524093531967, 5.90767299024270226404425939091, 6.39769483319444291966101608301, 7.01437951887825475719301902451, 8.001513662168442863978309623759