L(s) = 1 | − 0.725·5-s + 13.8·7-s + 20.3·11-s − 18.9·17-s − 19·19-s + 30·23-s − 24.4·25-s − 10.0·35-s + 53.8·43-s + 86.5·47-s + 142.·49-s − 14.7·55-s + 5.12·61-s − 112.·73-s + 281.·77-s − 90·83-s + 13.7·85-s + 13.7·95-s + 102·101-s − 21.7·115-s − 261.·119-s + ⋯ |
L(s) = 1 | − 0.145·5-s + 1.97·7-s + 1.85·11-s − 1.11·17-s − 19-s + 1.30·23-s − 0.978·25-s − 0.286·35-s + 1.25·43-s + 1.84·47-s + 2.90·49-s − 0.268·55-s + 0.0839·61-s − 1.53·73-s + 3.65·77-s − 1.08·83-s + 0.161·85-s + 0.145·95-s + 1.00·101-s − 0.189·115-s − 2.19·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.425828131\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.425828131\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 19T \) |
good | 5 | \( 1 + 0.725T + 25T^{2} \) |
| 7 | \( 1 - 13.8T + 49T^{2} \) |
| 11 | \( 1 - 20.3T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 + 18.9T + 289T^{2} \) |
| 23 | \( 1 - 30T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 53.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 86.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 5.12T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 112.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 90T + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.51649373564279431316514771391, −9.001838000421535897469975899857, −8.792489252675895982604013092641, −7.66845612593080650313103776332, −6.83628218871473453932384449871, −5.75699086513022957620937977201, −4.51925149592007014535183228697, −4.07649159717530113036768904413, −2.20236072010295605717780810898, −1.18835208932066842804317433537,
1.18835208932066842804317433537, 2.20236072010295605717780810898, 4.07649159717530113036768904413, 4.51925149592007014535183228697, 5.75699086513022957620937977201, 6.83628218871473453932384449871, 7.66845612593080650313103776332, 8.792489252675895982604013092641, 9.001838000421535897469975899857, 10.51649373564279431316514771391