| L(s) = 1 | + 4.47·7-s + 2.23·11-s − 4·13-s − 7·17-s + 6.70·19-s + 4.47·23-s − 4.47·31-s − 2·37-s − 5·41-s + 8.94·47-s + 13.0·49-s + 6·53-s + 8.94·59-s + 10·61-s − 2.23·67-s − 8.94·71-s + 9·73-s + 10.0·77-s + 4.47·79-s + 11.1·83-s + 5·89-s − 17.8·91-s − 2·97-s + 2·101-s − 8.94·103-s − 2.23·107-s + 6·109-s + ⋯ |
| L(s) = 1 | + 1.69·7-s + 0.674·11-s − 1.10·13-s − 1.69·17-s + 1.53·19-s + 0.932·23-s − 0.803·31-s − 0.328·37-s − 0.780·41-s + 1.30·47-s + 1.85·49-s + 0.824·53-s + 1.16·59-s + 1.28·61-s − 0.273·67-s − 1.06·71-s + 1.05·73-s + 1.13·77-s + 0.503·79-s + 1.22·83-s + 0.529·89-s − 1.87·91-s − 0.203·97-s + 0.199·101-s − 0.881·103-s − 0.216·107-s + 0.574·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.563699214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.563699214\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 7T + 17T^{2} \) |
| 19 | \( 1 - 6.70T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4.47T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 8.94T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 2.23T + 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 5T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−7.85919722703204252142985214027, −7.18074966830333952417161731756, −6.82523005268941573822036078649, −5.55841404155186887426314219062, −5.07372910773442354386177530480, −4.49333338076843366739838968674, −3.68136935289609968460499215237, −2.50248080542764954287182339896, −1.84646614113907720379604568650, −0.838075113007924729724253524185,
0.838075113007924729724253524185, 1.84646614113907720379604568650, 2.50248080542764954287182339896, 3.68136935289609968460499215237, 4.49333338076843366739838968674, 5.07372910773442354386177530480, 5.55841404155186887426314219062, 6.82523005268941573822036078649, 7.18074966830333952417161731756, 7.85919722703204252142985214027
