L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 12-s − 4·13-s + 16-s + 6·17-s − 18-s + 4·23-s + 24-s − 5·25-s + 4·26-s − 27-s + 6·29-s − 32-s − 6·34-s + 36-s − 4·37-s + 4·39-s − 10·41-s + 10·43-s − 4·46-s + 8·47-s − 48-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.288·12-s − 1.10·13-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.834·23-s + 0.204·24-s − 25-s + 0.784·26-s − 0.192·27-s + 1.11·29-s − 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.657·37-s + 0.640·39-s − 1.56·41-s + 1.52·43-s − 0.589·46-s + 1.16·47-s − 0.144·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8814767831\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8814767831\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−10.06559257649550112773931344727, −9.297128212952392596854709520982, −8.265244470170908999625864864578, −7.45414382667316107040197945820, −6.77687582776495632977992898709, −5.68332496275983467568160156186, −4.95371431860215373968668901127, −3.58313952852527998940732342675, −2.30840201774636776660554924947, −0.839344827324508039770418455749,
0.839344827324508039770418455749, 2.30840201774636776660554924947, 3.58313952852527998940732342675, 4.95371431860215373968668901127, 5.68332496275983467568160156186, 6.77687582776495632977992898709, 7.45414382667316107040197945820, 8.265244470170908999625864864578, 9.297128212952392596854709520982, 10.06559257649550112773931344727