L(s) = 1 | + 2.18·3-s + 5-s + 1.76·9-s + 3.59·11-s + 5.85·13-s + 2.18·15-s − 6.36·17-s + 4.50·19-s + 2.62·23-s + 25-s − 2.68·27-s + 2.05·29-s − 6.62·31-s + 7.85·33-s − 5.91·37-s + 12.7·39-s + 7.22·41-s − 5.34·43-s + 1.76·45-s + 2.31·47-s − 13.8·51-s − 4.10·53-s + 3.59·55-s + 9.83·57-s − 5.07·59-s − 7.37·61-s + 5.85·65-s + ⋯ |
L(s) = 1 | + 1.26·3-s + 0.447·5-s + 0.589·9-s + 1.08·11-s + 1.62·13-s + 0.563·15-s − 1.54·17-s + 1.03·19-s + 0.548·23-s + 0.200·25-s − 0.517·27-s + 0.382·29-s − 1.19·31-s + 1.36·33-s − 0.972·37-s + 2.04·39-s + 1.12·41-s − 0.815·43-s + 0.263·45-s + 0.338·47-s − 1.94·51-s − 0.564·53-s + 0.485·55-s + 1.30·57-s − 0.660·59-s − 0.944·61-s + 0.726·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.281743689\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.281743689\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 3 | \( 1 - 2.18T + 3T^{2} \) |
| 11 | \( 1 - 3.59T + 11T^{2} \) |
| 13 | \( 1 - 5.85T + 13T^{2} \) |
| 17 | \( 1 + 6.36T + 17T^{2} \) |
| 19 | \( 1 - 4.50T + 19T^{2} \) |
| 23 | \( 1 - 2.62T + 23T^{2} \) |
| 29 | \( 1 - 2.05T + 29T^{2} \) |
| 31 | \( 1 + 6.62T + 31T^{2} \) |
| 37 | \( 1 + 5.91T + 37T^{2} \) |
| 41 | \( 1 - 7.22T + 41T^{2} \) |
| 43 | \( 1 + 5.34T + 43T^{2} \) |
| 47 | \( 1 - 2.31T + 47T^{2} \) |
| 53 | \( 1 + 4.10T + 53T^{2} \) |
| 59 | \( 1 + 5.07T + 59T^{2} \) |
| 61 | \( 1 + 7.37T + 61T^{2} \) |
| 67 | \( 1 - 3.39T + 67T^{2} \) |
| 71 | \( 1 - 16.7T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 + 3.25T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−9.155131193804386228026010445300, −8.621844522313823097114848086414, −7.79372128415090563260377627848, −6.78588448829994298669180890006, −6.21101113640725637933791823883, −5.08643217852243509721350705530, −3.88812825835473333307733620093, −3.39197008299875536973001812958, −2.24617518121900846534714467486, −1.30935426042833557702822533132,
1.30935426042833557702822533132, 2.24617518121900846534714467486, 3.39197008299875536973001812958, 3.88812825835473333307733620093, 5.08643217852243509721350705530, 6.21101113640725637933791823883, 6.78588448829994298669180890006, 7.79372128415090563260377627848, 8.621844522313823097114848086414, 9.155131193804386228026010445300