| L(s) = 1 | + 0.332·3-s + 5-s + 0.507·7-s − 2.88·9-s − 0.290·11-s − 5.96·13-s + 0.332·15-s − 2.16·17-s + 8.41·19-s + 0.168·21-s − 3.58·23-s + 25-s − 1.95·27-s + 0.682·29-s + 4.02·31-s − 0.0965·33-s + 0.507·35-s − 5.47·37-s − 1.98·39-s − 9.91·41-s + 12.1·43-s − 2.88·45-s + 6.17·47-s − 6.74·49-s − 0.718·51-s + 10.7·53-s − 0.290·55-s + ⋯ |
| L(s) = 1 | + 0.191·3-s + 0.447·5-s + 0.191·7-s − 0.963·9-s − 0.0876·11-s − 1.65·13-s + 0.0857·15-s − 0.524·17-s + 1.93·19-s + 0.0367·21-s − 0.746·23-s + 0.200·25-s − 0.376·27-s + 0.126·29-s + 0.722·31-s − 0.0168·33-s + 0.0857·35-s − 0.899·37-s − 0.317·39-s − 1.54·41-s + 1.84·43-s − 0.430·45-s + 0.900·47-s − 0.963·49-s − 0.100·51-s + 1.47·53-s − 0.0391·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.802808113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.802808113\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 0.332T + 3T^{2} \) |
| 7 | \( 1 - 0.507T + 7T^{2} \) |
| 11 | \( 1 + 0.290T + 11T^{2} \) |
| 13 | \( 1 + 5.96T + 13T^{2} \) |
| 17 | \( 1 + 2.16T + 17T^{2} \) |
| 19 | \( 1 - 8.41T + 19T^{2} \) |
| 23 | \( 1 + 3.58T + 23T^{2} \) |
| 29 | \( 1 - 0.682T + 29T^{2} \) |
| 31 | \( 1 - 4.02T + 31T^{2} \) |
| 37 | \( 1 + 5.47T + 37T^{2} \) |
| 41 | \( 1 + 9.91T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 - 6.17T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 5.15T + 59T^{2} \) |
| 61 | \( 1 + 4.13T + 61T^{2} \) |
| 67 | \( 1 - 9.13T + 67T^{2} \) |
| 71 | \( 1 - 7.62T + 71T^{2} \) |
| 73 | \( 1 + 1.80T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 5.72T + 83T^{2} \) |
| 89 | \( 1 + 9.17T + 89T^{2} \) |
| 97 | \( 1 - 5.10T + 97T^{2} \) |
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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
−7.75299757218010987937023141152, −7.25219840126534590899990384485, −6.49009380175460805435490532805, −5.48930618653986054118652266936, −5.27437078043854578214879988152, −4.37572643268648011355477300289, −3.32529020694091327723893824140, −2.62894737550868036454456724940, −1.97771211561311141166628312891, −0.63297208417599048978502144731,
0.63297208417599048978502144731, 1.97771211561311141166628312891, 2.62894737550868036454456724940, 3.32529020694091327723893824140, 4.37572643268648011355477300289, 5.27437078043854578214879988152, 5.48930618653986054118652266936, 6.49009380175460805435490532805, 7.25219840126534590899990384485, 7.75299757218010987937023141152
