| L(s) = 1 | + 3-s + 4.37·5-s − 2.37·7-s + 9-s − 4.37·11-s + 2·13-s + 4.37·15-s − 4.37·17-s + 19-s − 2.37·21-s + 2.74·23-s + 14.1·25-s + 27-s − 8.74·29-s + 4.74·31-s − 4.37·33-s − 10.3·35-s − 6.74·37-s + 2·39-s − 2.37·43-s + 4.37·45-s − 7.62·47-s − 1.37·49-s − 4.37·51-s + 8.74·53-s − 19.1·55-s + 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.95·5-s − 0.896·7-s + 0.333·9-s − 1.31·11-s + 0.554·13-s + 1.12·15-s − 1.06·17-s + 0.229·19-s − 0.517·21-s + 0.572·23-s + 2.82·25-s + 0.192·27-s − 1.62·29-s + 0.852·31-s − 0.761·33-s − 1.75·35-s − 1.10·37-s + 0.320·39-s − 0.361·43-s + 0.651·45-s − 1.11·47-s − 0.196·49-s − 0.612·51-s + 1.20·53-s − 2.57·55-s + 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.691281738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.691281738\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 4.37T + 5T^{2} \) |
| 7 | \( 1 + 2.37T + 7T^{2} \) |
| 11 | \( 1 + 4.37T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 4.37T + 17T^{2} \) |
| 23 | \( 1 - 2.74T + 23T^{2} \) |
| 29 | \( 1 + 8.74T + 29T^{2} \) |
| 31 | \( 1 - 4.74T + 31T^{2} \) |
| 37 | \( 1 + 6.74T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 2.37T + 43T^{2} \) |
| 47 | \( 1 + 7.62T + 47T^{2} \) |
| 53 | \( 1 - 8.74T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8.37T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 0.372T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 2.74T + 83T^{2} \) |
| 89 | \( 1 - 3.25T + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−12.74495527959745433790785591282, −10.91566874243973867331741815141, −10.10473405641649195299306455158, −9.403901218778821671252801888359, −8.564694369666169277933712787216, −7.01685295635479423682123436759, −6.07571501726620236795664103014, −5.07471947894140600627574066538, −3.11087682315179490099576488506, −2.03234314274610698451890866583,
2.03234314274610698451890866583, 3.11087682315179490099576488506, 5.07471947894140600627574066538, 6.07571501726620236795664103014, 7.01685295635479423682123436759, 8.564694369666169277933712787216, 9.403901218778821671252801888359, 10.10473405641649195299306455158, 10.91566874243973867331741815141, 12.74495527959745433790785591282
