L(s) = 1 | + 2-s − 1.80·3-s + 4-s − 1.80·6-s + 8-s + 2.24·9-s − 0.445·11-s − 1.80·12-s + 16-s − 0.445·17-s + 2.24·18-s + 1.24·19-s − 0.445·22-s − 1.80·24-s + 25-s − 2.24·27-s + 32-s + 0.801·33-s − 0.445·34-s + 2.24·36-s + 1.24·38-s + 1.24·41-s + 1.24·43-s − 0.445·44-s − 1.80·48-s + 49-s + 50-s + ⋯ |
L(s) = 1 | + 2-s − 1.80·3-s + 4-s − 1.80·6-s + 8-s + 2.24·9-s − 0.445·11-s − 1.80·12-s + 16-s − 0.445·17-s + 2.24·18-s + 1.24·19-s − 0.445·22-s − 1.80·24-s + 25-s − 2.24·27-s + 32-s + 0.801·33-s − 0.445·34-s + 2.24·36-s + 1.24·38-s + 1.24·41-s + 1.24·43-s − 0.445·44-s − 1.80·48-s + 49-s + 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.218224212\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.218224212\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 1.80T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 0.445T + T^{2} \) |
| 17 | \( 1 + 0.445T + T^{2} \) |
| 19 | \( 1 - 1.24T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.24T + T^{2} \) |
| 43 | \( 1 - 1.24T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.80T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.80T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.80T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 1.24T + T^{2} \) |
| 89 | \( 1 + 1.80T + T^{2} \) |
| 97 | \( 1 + 0.445T + T^{2} \) |
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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
−10.31622996949810291538832858526, −9.215949014375014229922164153859, −7.62823286732801751367162785335, −7.11971171534175794957419728522, −6.16892165490899244271613806479, −5.63696719239523866602683037939, −4.86621300700599188181954170401, −4.19083423697713425188602266253, −2.81500047103094009942156870388, −1.23776052890527165988518872093,
1.23776052890527165988518872093, 2.81500047103094009942156870388, 4.19083423697713425188602266253, 4.86621300700599188181954170401, 5.63696719239523866602683037939, 6.16892165490899244271613806479, 7.11971171534175794957419728522, 7.62823286732801751367162785335, 9.215949014375014229922164153859, 10.31622996949810291538832858526