L(s) = 1 | + 2.20·3-s − 0.810·5-s − 4.21·7-s + 1.88·9-s − 4.98·11-s + 0.624·13-s − 1.79·15-s − 2.63·17-s + 4.79·19-s − 9.32·21-s + 3.72·23-s − 4.34·25-s − 2.46·27-s − 7.24·29-s + 6.65·31-s − 11.0·33-s + 3.41·35-s + 8.59·37-s + 1.37·39-s − 2.64·41-s − 0.862·43-s − 1.52·45-s + 0.217·47-s + 10.7·49-s − 5.82·51-s − 5.03·53-s + 4.03·55-s + ⋯ |
L(s) = 1 | + 1.27·3-s − 0.362·5-s − 1.59·7-s + 0.627·9-s − 1.50·11-s + 0.173·13-s − 0.462·15-s − 0.639·17-s + 1.10·19-s − 2.03·21-s + 0.777·23-s − 0.868·25-s − 0.474·27-s − 1.34·29-s + 1.19·31-s − 1.91·33-s + 0.578·35-s + 1.41·37-s + 0.220·39-s − 0.413·41-s − 0.131·43-s − 0.227·45-s + 0.0316·47-s + 1.54·49-s − 0.815·51-s − 0.691·53-s + 0.544·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.646725643\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.646725643\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 - 2.20T + 3T^{2} \) |
| 5 | \( 1 + 0.810T + 5T^{2} \) |
| 7 | \( 1 + 4.21T + 7T^{2} \) |
| 11 | \( 1 + 4.98T + 11T^{2} \) |
| 13 | \( 1 - 0.624T + 13T^{2} \) |
| 17 | \( 1 + 2.63T + 17T^{2} \) |
| 19 | \( 1 - 4.79T + 19T^{2} \) |
| 23 | \( 1 - 3.72T + 23T^{2} \) |
| 29 | \( 1 + 7.24T + 29T^{2} \) |
| 31 | \( 1 - 6.65T + 31T^{2} \) |
| 37 | \( 1 - 8.59T + 37T^{2} \) |
| 41 | \( 1 + 2.64T + 41T^{2} \) |
| 43 | \( 1 + 0.862T + 43T^{2} \) |
| 47 | \( 1 - 0.217T + 47T^{2} \) |
| 53 | \( 1 + 5.03T + 53T^{2} \) |
| 59 | \( 1 - 0.0696T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 8.17T + 71T^{2} \) |
| 73 | \( 1 + 0.949T + 73T^{2} \) |
| 79 | \( 1 + 6.08T + 79T^{2} \) |
| 83 | \( 1 - 1.19T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 2.98T + 97T^{2} \) |
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
−7.80293647852482070697540163842, −7.38910143446799205662316182646, −6.56727036824806661224314993450, −5.79304363885466113364727301237, −5.01252603337917631501945440890, −3.94055989812420094658122170935, −3.35220815820814343297032590150, −2.79592816083262135332944535990, −2.17754766930815812995423748739, −0.55549462108227411406038659240,
0.55549462108227411406038659240, 2.17754766930815812995423748739, 2.79592816083262135332944535990, 3.35220815820814343297032590150, 3.94055989812420094658122170935, 5.01252603337917631501945440890, 5.79304363885466113364727301237, 6.56727036824806661224314993450, 7.38910143446799205662316182646, 7.80293647852482070697540163842