L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 5·11-s + 13-s + 14-s + 16-s + 5·17-s + 5·22-s + 26-s + 28-s + 7·29-s − 9·31-s + 32-s + 5·34-s + 8·37-s + 2·41-s − 8·43-s + 5·44-s − 9·47-s − 6·49-s + 52-s + 11·53-s + 56-s + 7·58-s − 59-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 1.50·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 1.21·17-s + 1.06·22-s + 0.196·26-s + 0.188·28-s + 1.29·29-s − 1.61·31-s + 0.176·32-s + 0.857·34-s + 1.31·37-s + 0.312·41-s − 1.21·43-s + 0.753·44-s − 1.31·47-s − 6/7·49-s + 0.138·52-s + 1.51·53-s + 0.133·56-s + 0.919·58-s − 0.130·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.113594871\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.113594871\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.088492266040265701579949174860, −7.22612210193618057573870117837, −6.60318760116457776588301828871, −5.93434719081907500309953313384, −5.21077042412009560629832605549, −4.40199967862772452595324246988, −3.71098885793032632862471707203, −3.03564107829344958034836631456, −1.82168651604533637736463234721, −1.05997990870137429478328327500,
1.05997990870137429478328327500, 1.82168651604533637736463234721, 3.03564107829344958034836631456, 3.71098885793032632862471707203, 4.40199967862772452595324246988, 5.21077042412009560629832605549, 5.93434719081907500309953313384, 6.60318760116457776588301828871, 7.22612210193618057573870117837, 8.088492266040265701579949174860