L(s) = 1 | − 3.46·5-s + 1.41·7-s + 4.89·11-s − 1.41·13-s + 4.89·17-s + 6·19-s − 6.92·23-s + 6.99·25-s + 3.46·29-s − 1.41·31-s − 4.89·35-s + 9.89·37-s + 4.89·41-s + 6·43-s − 6.92·47-s − 5·49-s − 3.46·53-s − 16.9·55-s + 9.79·59-s + 7.07·61-s + 4.89·65-s − 8·67-s − 13.8·71-s − 12·73-s + 6.92·77-s + 15.5·79-s − 14.6·83-s + ⋯ |
L(s) = 1 | − 1.54·5-s + 0.534·7-s + 1.47·11-s − 0.392·13-s + 1.18·17-s + 1.37·19-s − 1.44·23-s + 1.39·25-s + 0.643·29-s − 0.254·31-s − 0.828·35-s + 1.62·37-s + 0.765·41-s + 0.914·43-s − 1.01·47-s − 0.714·49-s − 0.475·53-s − 2.28·55-s + 1.27·59-s + 0.905·61-s + 0.607·65-s − 0.977·67-s − 1.64·71-s − 1.40·73-s + 0.789·77-s + 1.75·79-s − 1.61·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.855206173\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.855206173\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 - 9.89T + 37T^{2} \) |
| 41 | \( 1 - 4.89T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 3.46T + 53T^{2} \) |
| 59 | \( 1 - 9.79T + 59T^{2} \) |
| 61 | \( 1 - 7.07T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 12T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 9.79T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−7.68853659041334032555242131890, −7.33398870348761309710734017153, −6.38794985598501149583836397751, −5.68759910319884912928512127127, −4.73778748247084350393802217505, −4.15371331839441195151583207274, −3.60443294835984048565874149679, −2.83294256647400753416577295156, −1.49494183456744791130164007006, −0.71552529136574057574932915843,
0.71552529136574057574932915843, 1.49494183456744791130164007006, 2.83294256647400753416577295156, 3.60443294835984048565874149679, 4.15371331839441195151583207274, 4.73778748247084350393802217505, 5.68759910319884912928512127127, 6.38794985598501149583836397751, 7.33398870348761309710734017153, 7.68853659041334032555242131890