| L(s) = 1 | + 0.848·3-s − 0.844·5-s + 7-s − 2.28·9-s − 11-s + 13-s − 0.716·15-s + 3.07·17-s + 2.28·19-s + 0.848·21-s + 0.257·23-s − 4.28·25-s − 4.47·27-s − 3.31·29-s + 8.43·31-s − 0.848·33-s − 0.844·35-s + 8.51·37-s + 0.848·39-s − 11.5·41-s + 10.8·43-s + 1.92·45-s + 2.60·47-s + 49-s + 2.61·51-s + 7.20·53-s + 0.844·55-s + ⋯ |
| L(s) = 1 | + 0.489·3-s − 0.377·5-s + 0.377·7-s − 0.760·9-s − 0.301·11-s + 0.277·13-s − 0.184·15-s + 0.746·17-s + 0.524·19-s + 0.185·21-s + 0.0536·23-s − 0.857·25-s − 0.862·27-s − 0.614·29-s + 1.51·31-s − 0.147·33-s − 0.142·35-s + 1.39·37-s + 0.135·39-s − 1.81·41-s + 1.65·43-s + 0.286·45-s + 0.379·47-s + 0.142·49-s + 0.365·51-s + 0.989·53-s + 0.113·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.008018925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.008018925\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 0.848T + 3T^{2} \) |
| 5 | \( 1 + 0.844T + 5T^{2} \) |
| 17 | \( 1 - 3.07T + 17T^{2} \) |
| 19 | \( 1 - 2.28T + 19T^{2} \) |
| 23 | \( 1 - 0.257T + 23T^{2} \) |
| 29 | \( 1 + 3.31T + 29T^{2} \) |
| 31 | \( 1 - 8.43T + 31T^{2} \) |
| 37 | \( 1 - 8.51T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 2.60T + 47T^{2} \) |
| 53 | \( 1 - 7.20T + 53T^{2} \) |
| 59 | \( 1 + 2.61T + 59T^{2} \) |
| 61 | \( 1 - 5.58T + 61T^{2} \) |
| 67 | \( 1 + 9.94T + 67T^{2} \) |
| 71 | \( 1 - 4.02T + 71T^{2} \) |
| 73 | \( 1 + 3.66T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + 2.21T + 83T^{2} \) |
| 89 | \( 1 - 8.87T + 89T^{2} \) |
| 97 | \( 1 - 9.92T + 97T^{2} \) |
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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
−8.278481730250604778330225017830, −7.888624249621010134120590024565, −7.22666092054875364869417199986, −6.08883694722323601751704525504, −5.55787838747093747384492239308, −4.62160785912038213090674841369, −3.72876516904486353491501601434, −2.99295163989557297627759944518, −2.11270213275765651641371274531, −0.790852294047762591475101495691,
0.790852294047762591475101495691, 2.11270213275765651641371274531, 2.99295163989557297627759944518, 3.72876516904486353491501601434, 4.62160785912038213090674841369, 5.55787838747093747384492239308, 6.08883694722323601751704525504, 7.22666092054875364869417199986, 7.888624249621010134120590024565, 8.278481730250604778330225017830
