| L(s) = 1 | + 3.62·2-s + 5.15·4-s − 22.4·7-s − 10.3·8-s + 52.4·11-s + 84.8·13-s − 81.4·14-s − 78.6·16-s + 45.4·17-s − 85.6·19-s + 190.·22-s + 55.7·23-s + 307.·26-s − 115.·28-s + 96.5·29-s + 107.·31-s − 202.·32-s + 164.·34-s + 81.7·37-s − 310.·38-s + 248.·41-s + 292.·43-s + 270.·44-s + 202.·46-s + 240.·47-s + 161.·49-s + 437.·52-s + ⋯ |
| L(s) = 1 | + 1.28·2-s + 0.644·4-s − 1.21·7-s − 0.456·8-s + 1.43·11-s + 1.80·13-s − 1.55·14-s − 1.22·16-s + 0.648·17-s − 1.03·19-s + 1.84·22-s + 0.505·23-s + 2.32·26-s − 0.781·28-s + 0.618·29-s + 0.623·31-s − 1.12·32-s + 0.831·34-s + 0.363·37-s − 1.32·38-s + 0.944·41-s + 1.03·43-s + 0.926·44-s + 0.648·46-s + 0.745·47-s + 0.469·49-s + 1.16·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.756374237\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.756374237\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 3.62T + 8T^{2} \) |
| 7 | \( 1 + 22.4T + 343T^{2} \) |
| 11 | \( 1 - 52.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 84.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 45.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 85.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 55.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 96.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 107.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 81.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 248.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 292.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 240.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 505.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 613.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 147.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 55.7T + 3.00e5T^{2} \) |
| 71 | \( 1 + 541.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 73.4T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.22e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 718.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 540.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.66e3T + 9.12e5T^{2} \) |
| 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
−10.18242778149750069190981347643, −9.116363056674675690304961296004, −8.609046428472292373505186857656, −6.94557305471801436576694000706, −6.24301128618410853576684274985, −5.78463938833146217880924987324, −4.21346226069852039513616982539, −3.77437566675857213201718462891, −2.77619783007338967401309349384, −0.973003085835479335974565493895,
0.973003085835479335974565493895, 2.77619783007338967401309349384, 3.77437566675857213201718462891, 4.21346226069852039513616982539, 5.78463938833146217880924987324, 6.24301128618410853576684274985, 6.94557305471801436576694000706, 8.609046428472292373505186857656, 9.116363056674675690304961296004, 10.18242778149750069190981347643
