| L(s) = 1 | + 2-s − 2.46·3-s + 4-s + 3.19·5-s − 2.46·6-s + 7-s + 8-s + 3.07·9-s + 3.19·10-s − 1.63·11-s − 2.46·12-s + 14-s − 7.87·15-s + 16-s − 3.38·17-s + 3.07·18-s + 5.90·19-s + 3.19·20-s − 2.46·21-s − 1.63·22-s + 6.07·23-s − 2.46·24-s + 5.19·25-s − 0.179·27-s + 28-s − 0.769·29-s − 7.87·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.42·3-s + 0.5·4-s + 1.42·5-s − 1.00·6-s + 0.377·7-s + 0.353·8-s + 1.02·9-s + 1.00·10-s − 0.494·11-s − 0.711·12-s + 0.267·14-s − 2.03·15-s + 0.250·16-s − 0.821·17-s + 0.724·18-s + 1.35·19-s + 0.714·20-s − 0.537·21-s − 0.349·22-s + 1.26·23-s − 0.503·24-s + 1.03·25-s − 0.0346·27-s + 0.188·28-s − 0.142·29-s − 1.43·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.408320358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.408320358\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2.46T + 3T^{2} \) |
| 5 | \( 1 - 3.19T + 5T^{2} \) |
| 11 | \( 1 + 1.63T + 11T^{2} \) |
| 17 | \( 1 + 3.38T + 17T^{2} \) |
| 19 | \( 1 - 5.90T + 19T^{2} \) |
| 23 | \( 1 - 6.07T + 23T^{2} \) |
| 29 | \( 1 + 0.769T + 29T^{2} \) |
| 31 | \( 1 - 8.79T + 31T^{2} \) |
| 37 | \( 1 + 8.07T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 8.70T + 47T^{2} \) |
| 53 | \( 1 - 3.75T + 53T^{2} \) |
| 59 | \( 1 + 8.61T + 59T^{2} \) |
| 61 | \( 1 - 1.62T + 61T^{2} \) |
| 67 | \( 1 - 6.49T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 - 4.98T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 + 7.70T + 97T^{2} \) |
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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
−9.193500479577136614013264343096, −8.089253414026761995792639738003, −6.86269718272050470917086671701, −6.56896139108696923454467914145, −5.58427737571701300688071229581, −5.21175521858518578131523608860, −4.66687029913061056566123996472, −3.17537953042340514038584364835, −2.10909702703217107909254216424, −1.01364058095720175265381653244,
1.01364058095720175265381653244, 2.10909702703217107909254216424, 3.17537953042340514038584364835, 4.66687029913061056566123996472, 5.21175521858518578131523608860, 5.58427737571701300688071229581, 6.56896139108696923454467914145, 6.86269718272050470917086671701, 8.089253414026761995792639738003, 9.193500479577136614013264343096
