| L(s) = 1 | + 3-s − 3.37·7-s + 9-s − 3.37·11-s + 13-s + 1.37·17-s − 6.74·19-s − 3.37·21-s + 0.627·23-s + 27-s − 2·29-s + 6.74·31-s − 3.37·33-s − 5.37·37-s + 39-s − 1.37·41-s + 4·43-s + 1.25·47-s + 4.37·49-s + 1.37·51-s + 9.37·53-s − 6.74·57-s − 8·59-s + 8.11·61-s − 3.37·63-s + 4·67-s + 0.627·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.27·7-s + 0.333·9-s − 1.01·11-s + 0.277·13-s + 0.332·17-s − 1.54·19-s − 0.735·21-s + 0.130·23-s + 0.192·27-s − 0.371·29-s + 1.21·31-s − 0.587·33-s − 0.883·37-s + 0.160·39-s − 0.214·41-s + 0.609·43-s + 0.183·47-s + 0.624·49-s + 0.192·51-s + 1.28·53-s − 0.893·57-s − 1.04·59-s + 1.03·61-s − 0.424·63-s + 0.488·67-s + 0.0755·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.539296560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.539296560\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 3.37T + 7T^{2} \) |
| 11 | \( 1 + 3.37T + 11T^{2} \) |
| 17 | \( 1 - 1.37T + 17T^{2} \) |
| 19 | \( 1 + 6.74T + 19T^{2} \) |
| 23 | \( 1 - 0.627T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 6.74T + 31T^{2} \) |
| 37 | \( 1 + 5.37T + 37T^{2} \) |
| 41 | \( 1 + 1.37T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 1.25T + 47T^{2} \) |
| 53 | \( 1 - 9.37T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 8.11T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 + 16.8T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 2.62T + 97T^{2} \) |
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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
−7.900624781357501619446937000506, −7.17174841255261310782399165734, −6.48331909557200591400581491507, −5.91953161039758471381773777191, −5.02203152591025360802718577379, −4.15105669996790774837832677608, −3.42816947744027731662080802639, −2.73514078628817502446976078479, −2.01733105175674800279413274452, −0.56984213413136972495473219661,
0.56984213413136972495473219661, 2.01733105175674800279413274452, 2.73514078628817502446976078479, 3.42816947744027731662080802639, 4.15105669996790774837832677608, 5.02203152591025360802718577379, 5.91953161039758471381773777191, 6.48331909557200591400581491507, 7.17174841255261310782399165734, 7.900624781357501619446937000506
