L(s) = 1 | − 0.732·3-s + 2·7-s − 2.46·9-s − 3.46·11-s − 0.732·13-s − 3.46·17-s − 19-s − 1.46·21-s + 3.46·23-s + 4·27-s − 3.46·29-s − 5.46·31-s + 2.53·33-s − 3.26·37-s + 0.535·39-s − 6·41-s + 8.92·43-s − 0.928·47-s − 3·49-s + 2.53·51-s + 7.26·53-s + 0.732·57-s + 6.92·59-s − 8.39·61-s − 4.92·63-s + 3.26·67-s − 2.53·69-s + ⋯ |
L(s) = 1 | − 0.422·3-s + 0.755·7-s − 0.821·9-s − 1.04·11-s − 0.203·13-s − 0.840·17-s − 0.229·19-s − 0.319·21-s + 0.722·23-s + 0.769·27-s − 0.643·29-s − 0.981·31-s + 0.441·33-s − 0.537·37-s + 0.0858·39-s − 0.937·41-s + 1.36·43-s − 0.135·47-s − 0.428·49-s + 0.355·51-s + 0.998·53-s + 0.0969·57-s + 0.901·59-s − 1.07·61-s − 0.620·63-s + 0.399·67-s − 0.305·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.065366184\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.065366184\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 0.732T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 + 3.26T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8.92T + 43T^{2} \) |
| 47 | \( 1 + 0.928T + 47T^{2} \) |
| 53 | \( 1 - 7.26T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 + 8.39T + 61T^{2} \) |
| 67 | \( 1 - 3.26T + 67T^{2} \) |
| 71 | \( 1 - 9.46T + 71T^{2} \) |
| 73 | \( 1 - 7.46T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 + 8.53T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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
−7.889325726438903159238616725052, −7.20289814273498266109499431047, −6.49653568166326509078440693125, −5.54395953320593624690323702646, −5.23074115733347879785666970741, −4.50742078205795462240653246559, −3.51702835688572630748210485359, −2.58180902689360741566359779956, −1.88705711337291267248016446614, −0.50613423632576486396932258033,
0.50613423632576486396932258033, 1.88705711337291267248016446614, 2.58180902689360741566359779956, 3.51702835688572630748210485359, 4.50742078205795462240653246559, 5.23074115733347879785666970741, 5.54395953320593624690323702646, 6.49653568166326509078440693125, 7.20289814273498266109499431047, 7.889325726438903159238616725052