L(s) = 1 | + 1.87·3-s + 5-s + 7-s + 0.518·9-s + 1.08·11-s − 13-s + 1.87·15-s + 4.65·17-s + 1.78·19-s + 1.87·21-s − 9.27·23-s + 25-s − 4.65·27-s − 1.48·29-s − 2.96·31-s + 2.04·33-s + 35-s + 11.1·37-s − 1.87·39-s + 3.57·41-s + 11.4·43-s + 0.518·45-s − 7.40·47-s + 49-s + 8.72·51-s + 0.822·53-s + 1.08·55-s + ⋯ |
L(s) = 1 | + 1.08·3-s + 0.447·5-s + 0.377·7-s + 0.172·9-s + 0.328·11-s − 0.277·13-s + 0.484·15-s + 1.12·17-s + 0.409·19-s + 0.409·21-s − 1.93·23-s + 0.200·25-s − 0.895·27-s − 0.275·29-s − 0.532·31-s + 0.355·33-s + 0.169·35-s + 1.82·37-s − 0.300·39-s + 0.557·41-s + 1.74·43-s + 0.0772·45-s − 1.07·47-s + 0.142·49-s + 1.22·51-s + 0.112·53-s + 0.146·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.599449590\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.599449590\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 - 1.87T + 3T^{2} \) |
| 11 | \( 1 - 1.08T + 11T^{2} \) |
| 17 | \( 1 - 4.65T + 17T^{2} \) |
| 19 | \( 1 - 1.78T + 19T^{2} \) |
| 23 | \( 1 + 9.27T + 23T^{2} \) |
| 29 | \( 1 + 1.48T + 29T^{2} \) |
| 31 | \( 1 + 2.96T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 - 3.57T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 7.40T + 47T^{2} \) |
| 53 | \( 1 - 0.822T + 53T^{2} \) |
| 59 | \( 1 - 6.96T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 6.08T + 79T^{2} \) |
| 83 | \( 1 + 6.14T + 83T^{2} \) |
| 89 | \( 1 + 3.38T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−7.84419852710062439894958435019, −7.61576394596980191463493238107, −6.50040837944902149141157029219, −5.77783020968664583506811826208, −5.19127090522945427427660450736, −4.03566502393785862159576085430, −3.62230518967738777258982516675, −2.52592102701094239986994320777, −2.08475005368387795837463036969, −0.925479797481114343268151557976,
0.925479797481114343268151557976, 2.08475005368387795837463036969, 2.52592102701094239986994320777, 3.62230518967738777258982516675, 4.03566502393785862159576085430, 5.19127090522945427427660450736, 5.77783020968664583506811826208, 6.50040837944902149141157029219, 7.61576394596980191463493238107, 7.84419852710062439894958435019