| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 9-s + 3·11-s − 2·12-s + 5·13-s + 16-s + 6·17-s − 18-s + 19-s − 3·22-s − 3·23-s + 2·24-s − 5·26-s + 4·27-s − 6·29-s + 4·31-s − 32-s − 6·33-s − 6·34-s + 36-s − 11·37-s − 38-s − 10·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s + 0.904·11-s − 0.577·12-s + 1.38·13-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.229·19-s − 0.639·22-s − 0.625·23-s + 0.408·24-s − 0.980·26-s + 0.769·27-s − 1.11·29-s + 0.718·31-s − 0.176·32-s − 1.04·33-s − 1.02·34-s + 1/6·36-s − 1.80·37-s − 0.162·38-s − 1.60·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9220928838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9220928838\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−8.916910074221539449465817934597, −8.275790348636825325485669788414, −7.35175521207325832158499389696, −6.55889975531230708182956656690, −5.87603590927126677777617518210, −5.38825983921415985856014202819, −4.05886524291970267937333390052, −3.24918340725702523141175696293, −1.64611302323670967334868050239, −0.77109983016600816116103655159,
0.77109983016600816116103655159, 1.64611302323670967334868050239, 3.24918340725702523141175696293, 4.05886524291970267937333390052, 5.38825983921415985856014202819, 5.87603590927126677777617518210, 6.55889975531230708182956656690, 7.35175521207325832158499389696, 8.275790348636825325485669788414, 8.916910074221539449465817934597
