L(s) = 1 | + 2.56·2-s − 3·3-s − 1.43·4-s − 7.68·6-s + 29.1·7-s − 24.1·8-s + 9·9-s − 11·11-s + 4.31·12-s − 39.1·13-s + 74.5·14-s − 50.4·16-s + 46.3·17-s + 23.0·18-s + 43.5·19-s − 87.3·21-s − 28.1·22-s + 91.3·23-s + 72.5·24-s − 100.·26-s − 27·27-s − 41.8·28-s − 185.·29-s − 45.8·31-s + 64.2·32-s + 33·33-s + 118.·34-s + ⋯ |
L(s) = 1 | + 0.905·2-s − 0.577·3-s − 0.179·4-s − 0.522·6-s + 1.57·7-s − 1.06·8-s + 0.333·9-s − 0.301·11-s + 0.103·12-s − 0.835·13-s + 1.42·14-s − 0.787·16-s + 0.661·17-s + 0.301·18-s + 0.526·19-s − 0.907·21-s − 0.273·22-s + 0.827·23-s + 0.616·24-s − 0.756·26-s − 0.192·27-s − 0.282·28-s − 1.19·29-s − 0.265·31-s + 0.354·32-s + 0.174·33-s + 0.598·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.548621355\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.548621355\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
good | 2 | \( 1 - 2.56T + 8T^{2} \) |
| 7 | \( 1 - 29.1T + 343T^{2} \) |
| 13 | \( 1 + 39.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 46.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 43.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 91.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 185.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 45.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 177.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 232.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 64.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 16.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 48.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 200.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 521.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 338.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 318.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.13e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 824.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 731.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 902.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.35e3T + 9.12e5T^{2} \) |
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
−9.879549703216117587010066408154, −9.039928171162925809076004781042, −7.941778801968153104467781814079, −7.27729417442585314005623359407, −5.92239498824741225317861466140, −5.15752270681188957744070231407, −4.75221179719072633600753777248, −3.65786041916406184672783970382, −2.28132211055306567908884272537, −0.816261344864138847409443634594,
0.816261344864138847409443634594, 2.28132211055306567908884272537, 3.65786041916406184672783970382, 4.75221179719072633600753777248, 5.15752270681188957744070231407, 5.92239498824741225317861466140, 7.27729417442585314005623359407, 7.941778801968153104467781814079, 9.039928171162925809076004781042, 9.879549703216117587010066408154