| L(s) = 1 | + 0.676·3-s + 3.59·5-s + 7-s − 2.54·9-s − 11-s − 13-s + 2.43·15-s − 5.62·17-s − 5.78·19-s + 0.676·21-s + 8.70·23-s + 7.91·25-s − 3.75·27-s + 9.58·29-s + 3.11·31-s − 0.676·33-s + 3.59·35-s − 5.39·37-s − 0.676·39-s + 8.94·41-s − 4.02·43-s − 9.13·45-s + 1.80·47-s + 49-s − 3.80·51-s − 4.67·53-s − 3.59·55-s + ⋯ |
| L(s) = 1 | + 0.390·3-s + 1.60·5-s + 0.377·7-s − 0.847·9-s − 0.301·11-s − 0.277·13-s + 0.627·15-s − 1.36·17-s − 1.32·19-s + 0.147·21-s + 1.81·23-s + 1.58·25-s − 0.721·27-s + 1.78·29-s + 0.559·31-s − 0.117·33-s + 0.607·35-s − 0.887·37-s − 0.108·39-s + 1.39·41-s − 0.613·43-s − 1.36·45-s + 0.263·47-s + 0.142·49-s − 0.532·51-s − 0.641·53-s − 0.484·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.054166893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.054166893\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.676T + 3T^{2} \) |
| 5 | \( 1 - 3.59T + 5T^{2} \) |
| 17 | \( 1 + 5.62T + 17T^{2} \) |
| 19 | \( 1 + 5.78T + 19T^{2} \) |
| 23 | \( 1 - 8.70T + 23T^{2} \) |
| 29 | \( 1 - 9.58T + 29T^{2} \) |
| 31 | \( 1 - 3.11T + 31T^{2} \) |
| 37 | \( 1 + 5.39T + 37T^{2} \) |
| 41 | \( 1 - 8.94T + 41T^{2} \) |
| 43 | \( 1 + 4.02T + 43T^{2} \) |
| 47 | \( 1 - 1.80T + 47T^{2} \) |
| 53 | \( 1 + 4.67T + 53T^{2} \) |
| 59 | \( 1 - 3.19T + 59T^{2} \) |
| 61 | \( 1 - 9.66T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 4.18T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 0.112T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 16.7T + 97T^{2} \) |
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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
−8.091829027891878835588921392268, −6.80508532888065616482643566770, −6.58849559490355438711395657728, −5.75504902704903939318200039736, −5.02127556791168040894988646543, −4.54903752567611196440294925515, −3.25211220989511351791809790879, −2.32778293813320108339055355589, −2.21022460218798287254113095578, −0.833969023280542130836075593919,
0.833969023280542130836075593919, 2.21022460218798287254113095578, 2.32778293813320108339055355589, 3.25211220989511351791809790879, 4.54903752567611196440294925515, 5.02127556791168040894988646543, 5.75504902704903939318200039736, 6.58849559490355438711395657728, 6.80508532888065616482643566770, 8.091829027891878835588921392268
