| L(s) = 1 | − 0.290·3-s + 0.725·5-s − 7-s − 2.91·9-s + 11-s + 13-s − 0.210·15-s + 5.87·17-s + 6.34·19-s + 0.290·21-s − 1.90·23-s − 4.47·25-s + 1.71·27-s − 1.28·29-s + 6.23·31-s − 0.290·33-s − 0.725·35-s − 11.8·37-s − 0.290·39-s + 1.13·41-s − 3.11·43-s − 2.11·45-s − 1.90·47-s + 49-s − 1.70·51-s + 2.01·53-s + 0.725·55-s + ⋯ |
| L(s) = 1 | − 0.167·3-s + 0.324·5-s − 0.377·7-s − 0.971·9-s + 0.301·11-s + 0.277·13-s − 0.0543·15-s + 1.42·17-s + 1.45·19-s + 0.0633·21-s − 0.396·23-s − 0.894·25-s + 0.330·27-s − 0.238·29-s + 1.11·31-s − 0.0505·33-s − 0.122·35-s − 1.95·37-s − 0.0464·39-s + 0.177·41-s − 0.475·43-s − 0.315·45-s − 0.277·47-s + 0.142·49-s − 0.238·51-s + 0.276·53-s + 0.0977·55-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
| \(L(1)\) |
\(\approx\) |
\(1.720519585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.720519585\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.290T + 3T^{2} \) |
| 5 | \( 1 - 0.725T + 5T^{2} \) |
| 17 | \( 1 - 5.87T + 17T^{2} \) |
| 19 | \( 1 - 6.34T + 19T^{2} \) |
| 23 | \( 1 + 1.90T + 23T^{2} \) |
| 29 | \( 1 + 1.28T + 29T^{2} \) |
| 31 | \( 1 - 6.23T + 31T^{2} \) |
| 37 | \( 1 + 11.8T + 37T^{2} \) |
| 41 | \( 1 - 1.13T + 41T^{2} \) |
| 43 | \( 1 + 3.11T + 43T^{2} \) |
| 47 | \( 1 + 1.90T + 47T^{2} \) |
| 53 | \( 1 - 2.01T + 53T^{2} \) |
| 59 | \( 1 + 2.94T + 59T^{2} \) |
| 61 | \( 1 - 9.69T + 61T^{2} \) |
| 67 | \( 1 - 3.12T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 2.27T + 73T^{2} \) |
| 79 | \( 1 - 3.11T + 79T^{2} \) |
| 83 | \( 1 - 7.10T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−8.437412763000122527788030237996, −7.75424017524692834233714559076, −6.96837784296830439936630721401, −6.03139255487776397875092438234, −5.63519319555959391473111428191, −4.87171373078218481483011199615, −3.54382923405995946735790371014, −3.17958084405570639923645908270, −1.93950067949716787825757204639, −0.76334418696860147219945602977,
0.76334418696860147219945602977, 1.93950067949716787825757204639, 3.17958084405570639923645908270, 3.54382923405995946735790371014, 4.87171373078218481483011199615, 5.63519319555959391473111428191, 6.03139255487776397875092438234, 6.96837784296830439936630721401, 7.75424017524692834233714559076, 8.437412763000122527788030237996
