| L(s) = 1 | − 0.146·3-s − 5-s − 0.146·7-s − 2.97·9-s − 2.68·11-s − 13-s + 0.146·15-s + 17-s + 4·19-s + 0.0214·21-s − 6.68·23-s − 4·25-s + 0.875·27-s + 4.39·29-s − 1.31·31-s + 0.393·33-s + 0.146·35-s + 3.97·37-s + 0.146·39-s + 6.39·41-s + 6.83·43-s + 2.97·45-s + 7.12·47-s − 6.97·49-s − 0.146·51-s − 8.97·53-s + 2.68·55-s + ⋯ |
| L(s) = 1 | − 0.0845·3-s − 0.447·5-s − 0.0553·7-s − 0.992·9-s − 0.809·11-s − 0.277·13-s + 0.0377·15-s + 0.242·17-s + 0.917·19-s + 0.00467·21-s − 1.39·23-s − 0.800·25-s + 0.168·27-s + 0.815·29-s − 0.236·31-s + 0.0684·33-s + 0.0247·35-s + 0.654·37-s + 0.0234·39-s + 0.998·41-s + 1.04·43-s + 0.444·45-s + 1.03·47-s − 0.996·49-s − 0.0204·51-s − 1.23·53-s + 0.362·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.071686919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.071686919\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 0.146T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 + 0.146T + 7T^{2} \) |
| 11 | \( 1 + 2.68T + 11T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 6.68T + 23T^{2} \) |
| 29 | \( 1 - 4.39T + 29T^{2} \) |
| 31 | \( 1 + 1.31T + 31T^{2} \) |
| 37 | \( 1 - 3.97T + 37T^{2} \) |
| 41 | \( 1 - 6.39T + 41T^{2} \) |
| 43 | \( 1 - 6.83T + 43T^{2} \) |
| 47 | \( 1 - 7.12T + 47T^{2} \) |
| 53 | \( 1 + 8.97T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 + 8.35T + 61T^{2} \) |
| 67 | \( 1 - 8.29T + 67T^{2} \) |
| 71 | \( 1 - 5.51T + 71T^{2} \) |
| 73 | \( 1 - 6.97T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 - 4.29T + 83T^{2} \) |
| 89 | \( 1 - 5.37T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−8.437301549285348949297834462663, −7.88332104914207440562079557007, −7.37885660330791331174807553777, −6.18525958590514365242128576050, −5.68443935388885160703579171789, −4.84611143282581803109274869288, −3.90749768276522982218973305051, −3.00759361378393382040499509290, −2.20269429879951795135460986513, −0.59289271013422027632471689904,
0.59289271013422027632471689904, 2.20269429879951795135460986513, 3.00759361378393382040499509290, 3.90749768276522982218973305051, 4.84611143282581803109274869288, 5.68443935388885160703579171789, 6.18525958590514365242128576050, 7.37885660330791331174807553777, 7.88332104914207440562079557007, 8.437301549285348949297834462663
