| L(s) = 1 | − 2-s − 7·4-s + 7·7-s + 15·8-s + 8·11-s − 28·13-s − 7·14-s + 41·16-s + 54·17-s − 110·19-s − 8·22-s + 48·23-s + 28·26-s − 49·28-s + 110·29-s + 12·31-s − 161·32-s − 54·34-s + 246·37-s + 110·38-s − 182·41-s − 128·43-s − 56·44-s − 48·46-s + 324·47-s + 49·49-s + 196·52-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 7/8·4-s + 0.377·7-s + 0.662·8-s + 0.219·11-s − 0.597·13-s − 0.133·14-s + 0.640·16-s + 0.770·17-s − 1.32·19-s − 0.0775·22-s + 0.435·23-s + 0.211·26-s − 0.330·28-s + 0.704·29-s + 0.0695·31-s − 0.889·32-s − 0.272·34-s + 1.09·37-s + 0.469·38-s − 0.693·41-s − 0.453·43-s − 0.191·44-s − 0.153·46-s + 1.00·47-s + 1/7·49-s + 0.522·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.233180215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.233180215\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 11 | \( 1 - 8 T + p^{3} T^{2} \) |
| 13 | \( 1 + 28 T + p^{3} T^{2} \) |
| 17 | \( 1 - 54 T + p^{3} T^{2} \) |
| 19 | \( 1 + 110 T + p^{3} T^{2} \) |
| 23 | \( 1 - 48 T + p^{3} T^{2} \) |
| 29 | \( 1 - 110 T + p^{3} T^{2} \) |
| 31 | \( 1 - 12 T + p^{3} T^{2} \) |
| 37 | \( 1 - 246 T + p^{3} T^{2} \) |
| 41 | \( 1 + 182 T + p^{3} T^{2} \) |
| 43 | \( 1 + 128 T + p^{3} T^{2} \) |
| 47 | \( 1 - 324 T + p^{3} T^{2} \) |
| 53 | \( 1 + 162 T + p^{3} T^{2} \) |
| 59 | \( 1 + 810 T + p^{3} T^{2} \) |
| 61 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 67 | \( 1 + 244 T + p^{3} T^{2} \) |
| 71 | \( 1 - 768 T + p^{3} T^{2} \) |
| 73 | \( 1 - 702 T + p^{3} T^{2} \) |
| 79 | \( 1 - 440 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1302 T + p^{3} T^{2} \) |
| 89 | \( 1 + 730 T + p^{3} T^{2} \) |
| 97 | \( 1 + 294 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.067187072381051863608173318004, −8.282459476537823874629761173919, −7.72938005258393646852253756995, −6.72445314954249621410592366333, −5.71094810690371343328176161597, −4.78063556320611405529541362531, −4.19918326544852037676722854595, −3.01452537712459475460663088657, −1.70295640232551494075010748677, −0.58797355499025672965663833983,
0.58797355499025672965663833983, 1.70295640232551494075010748677, 3.01452537712459475460663088657, 4.19918326544852037676722854595, 4.78063556320611405529541362531, 5.71094810690371343328176161597, 6.72445314954249621410592366333, 7.72938005258393646852253756995, 8.282459476537823874629761173919, 9.067187072381051863608173318004
