L(s) = 1 | + 5.42·2-s + 3·3-s + 21.4·4-s − 20.0·5-s + 16.2·6-s + 33.9·7-s + 73.0·8-s + 9·9-s − 108.·10-s − 1.17·11-s + 64.3·12-s − 21.8·13-s + 184.·14-s − 60.1·15-s + 225.·16-s − 115.·17-s + 48.8·18-s + 8.20·19-s − 430.·20-s + 101.·21-s − 6.36·22-s + 44.0·23-s + 219.·24-s + 276.·25-s − 118.·26-s + 27·27-s + 728.·28-s + ⋯ |
L(s) = 1 | + 1.91·2-s + 0.577·3-s + 2.68·4-s − 1.79·5-s + 1.10·6-s + 1.83·7-s + 3.23·8-s + 0.333·9-s − 3.43·10-s − 0.0321·11-s + 1.54·12-s − 0.466·13-s + 3.51·14-s − 1.03·15-s + 3.51·16-s − 1.65·17-s + 0.639·18-s + 0.0990·19-s − 4.80·20-s + 1.05·21-s − 0.0616·22-s + 0.399·23-s + 1.86·24-s + 2.21·25-s − 0.894·26-s + 0.192·27-s + 4.91·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(5.793017694\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.793017694\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 67 | \( 1 - 67T \) |
good | 2 | \( 1 - 5.42T + 8T^{2} \) |
| 5 | \( 1 + 20.0T + 125T^{2} \) |
| 7 | \( 1 - 33.9T + 343T^{2} \) |
| 11 | \( 1 + 1.17T + 1.33e3T^{2} \) |
| 13 | \( 1 + 21.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 115.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 8.20T + 6.85e3T^{2} \) |
| 23 | \( 1 - 44.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 169.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 14.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 112.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 7.90T + 6.89e4T^{2} \) |
| 43 | \( 1 + 448.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 342.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 39.9T + 1.48e5T^{2} \) |
| 59 | \( 1 + 126.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 308.T + 2.26e5T^{2} \) |
| 71 | \( 1 - 1.09e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 225.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 31.5T + 4.93e5T^{2} \) |
| 83 | \( 1 + 514.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 873.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 309.T + 9.12e5T^{2} \) |
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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
−12.02090755567024226875328852472, −11.34208455693682143712183931252, −10.90867909528766619431832099583, −8.466822323971601035010196934307, −7.66584694259418470426444306978, −6.90380395612456598058860020028, −4.95942087290542895856727499995, −4.48273775840276456244405313332, −3.47013417978219319118816036044, −1.99518250845458040922406622160,
1.99518250845458040922406622160, 3.47013417978219319118816036044, 4.48273775840276456244405313332, 4.95942087290542895856727499995, 6.90380395612456598058860020028, 7.66584694259418470426444306978, 8.466822323971601035010196934307, 10.90867909528766619431832099583, 11.34208455693682143712183931252, 12.02090755567024226875328852472