| L(s) = 1 | + 2-s + 2.85·3-s + 4-s − 5-s + 2.85·6-s + 0.594·7-s + 8-s + 5.13·9-s − 10-s + 0.279·11-s + 2.85·12-s + 2.07·13-s + 0.594·14-s − 2.85·15-s + 16-s − 1.23·17-s + 5.13·18-s + 6.28·19-s − 20-s + 1.69·21-s + 0.279·22-s − 2.47·23-s + 2.85·24-s + 25-s + 2.07·26-s + 6.09·27-s + 0.594·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.64·3-s + 0.5·4-s − 0.447·5-s + 1.16·6-s + 0.224·7-s + 0.353·8-s + 1.71·9-s − 0.316·10-s + 0.0841·11-s + 0.823·12-s + 0.576·13-s + 0.158·14-s − 0.736·15-s + 0.250·16-s − 0.299·17-s + 1.21·18-s + 1.44·19-s − 0.223·20-s + 0.370·21-s + 0.0594·22-s − 0.516·23-s + 0.582·24-s + 0.200·25-s + 0.407·26-s + 1.17·27-s + 0.112·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.212697093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.212697093\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 - 2.85T + 3T^{2} \) |
| 7 | \( 1 - 0.594T + 7T^{2} \) |
| 11 | \( 1 - 0.279T + 11T^{2} \) |
| 13 | \( 1 - 2.07T + 13T^{2} \) |
| 17 | \( 1 + 1.23T + 17T^{2} \) |
| 19 | \( 1 - 6.28T + 19T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 - 0.543T + 29T^{2} \) |
| 31 | \( 1 - 6.85T + 31T^{2} \) |
| 37 | \( 1 + 4.29T + 37T^{2} \) |
| 41 | \( 1 - 5.90T + 41T^{2} \) |
| 43 | \( 1 + 6.48T + 43T^{2} \) |
| 47 | \( 1 + 1.46T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 3.48T + 59T^{2} \) |
| 61 | \( 1 + 8.85T + 61T^{2} \) |
| 67 | \( 1 - 8.21T + 67T^{2} \) |
| 71 | \( 1 - 5.69T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 2.46T + 79T^{2} \) |
| 83 | \( 1 + 1.11T + 83T^{2} \) |
| 89 | \( 1 + 2.06T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−8.109392294145513904958535024611, −7.46842935074286760159320212216, −6.84507572415514673571768362131, −5.94154843554020333076191182571, −4.94808308759009063342226971634, −4.24010661498515833333549154357, −3.50946216065806731057685150919, −3.02367682619373925391465029508, −2.14331883691849727235155928253, −1.19207432527379684481683187158,
1.19207432527379684481683187158, 2.14331883691849727235155928253, 3.02367682619373925391465029508, 3.50946216065806731057685150919, 4.24010661498515833333549154357, 4.94808308759009063342226971634, 5.94154843554020333076191182571, 6.84507572415514673571768362131, 7.46842935074286760159320212216, 8.109392294145513904958535024611
