| L(s) = 1 | − 2-s + 3.28·3-s + 4-s − 2.84·5-s − 3.28·6-s + 1.28·7-s − 8-s + 7.76·9-s + 2.84·10-s + 11-s + 3.28·12-s − 4.11·13-s − 1.28·14-s − 9.33·15-s + 16-s + 2.60·17-s − 7.76·18-s + 6.27·19-s − 2.84·20-s + 4.20·21-s − 22-s + 4.31·23-s − 3.28·24-s + 3.09·25-s + 4.11·26-s + 15.6·27-s + 1.28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.89·3-s + 0.5·4-s − 1.27·5-s − 1.33·6-s + 0.484·7-s − 0.353·8-s + 2.58·9-s + 0.899·10-s + 0.301·11-s + 0.947·12-s − 1.14·13-s − 0.342·14-s − 2.40·15-s + 0.250·16-s + 0.631·17-s − 1.83·18-s + 1.43·19-s − 0.636·20-s + 0.918·21-s − 0.213·22-s + 0.899·23-s − 0.669·24-s + 0.618·25-s + 0.807·26-s + 3.00·27-s + 0.242·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.576740859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.576740859\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 - 3.28T + 3T^{2} \) |
| 5 | \( 1 + 2.84T + 5T^{2} \) |
| 7 | \( 1 - 1.28T + 7T^{2} \) |
| 13 | \( 1 + 4.11T + 13T^{2} \) |
| 17 | \( 1 - 2.60T + 17T^{2} \) |
| 19 | \( 1 - 6.27T + 19T^{2} \) |
| 23 | \( 1 - 4.31T + 23T^{2} \) |
| 29 | \( 1 + 0.988T + 29T^{2} \) |
| 31 | \( 1 + 3.96T + 31T^{2} \) |
| 37 | \( 1 - 7.87T + 37T^{2} \) |
| 41 | \( 1 - 4.88T + 41T^{2} \) |
| 43 | \( 1 + 3.27T + 43T^{2} \) |
| 47 | \( 1 + 5.58T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 - 6.86T + 59T^{2} \) |
| 61 | \( 1 + 1.86T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 4.96T + 71T^{2} \) |
| 73 | \( 1 - 5.99T + 73T^{2} \) |
| 79 | \( 1 - 1.63T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 5.26T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−8.144454831810927096770544879183, −7.83886160716910088963688239545, −7.41470272355048190025519876822, −6.77389149033511301528433616515, −5.16991197576410467781969260699, −4.37506374718018790788420193168, −3.45546802294415922603057501758, −3.01559003978332927085731644267, −2.00230658937759783247601752492, −0.956793626444517700585747832505,
0.956793626444517700585747832505, 2.00230658937759783247601752492, 3.01559003978332927085731644267, 3.45546802294415922603057501758, 4.37506374718018790788420193168, 5.16991197576410467781969260699, 6.77389149033511301528433616515, 7.41470272355048190025519876822, 7.83886160716910088963688239545, 8.144454831810927096770544879183
