L(s) = 1 | − 0.718·2-s + 0.0658·3-s − 1.48·4-s − 0.0473·6-s + 2.64·7-s + 2.50·8-s − 2.99·9-s − 4.10·11-s − 0.0976·12-s − 5.95·13-s − 1.89·14-s + 1.16·16-s + 2.15·18-s + 7.10·19-s + 0.174·21-s + 2.95·22-s + 3.02·23-s + 0.164·24-s + 4.28·26-s − 0.394·27-s − 3.91·28-s + 7.45·29-s + 0.0419·31-s − 5.84·32-s − 0.270·33-s + 4.44·36-s − 10.8·37-s + ⋯ |
L(s) = 1 | − 0.508·2-s + 0.0380·3-s − 0.741·4-s − 0.0193·6-s + 0.998·7-s + 0.885·8-s − 0.998·9-s − 1.23·11-s − 0.0281·12-s − 1.65·13-s − 0.507·14-s + 0.291·16-s + 0.507·18-s + 1.63·19-s + 0.0379·21-s + 0.629·22-s + 0.629·23-s + 0.0336·24-s + 0.840·26-s − 0.0759·27-s − 0.740·28-s + 1.38·29-s + 0.00753·31-s − 1.03·32-s − 0.0471·33-s + 0.740·36-s − 1.77·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7567637595\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7567637595\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + 0.718T + 2T^{2} \) |
| 3 | \( 1 - 0.0658T + 3T^{2} \) |
| 7 | \( 1 - 2.64T + 7T^{2} \) |
| 11 | \( 1 + 4.10T + 11T^{2} \) |
| 13 | \( 1 + 5.95T + 13T^{2} \) |
| 19 | \( 1 - 7.10T + 19T^{2} \) |
| 23 | \( 1 - 3.02T + 23T^{2} \) |
| 29 | \( 1 - 7.45T + 29T^{2} \) |
| 31 | \( 1 - 0.0419T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 9.07T + 41T^{2} \) |
| 43 | \( 1 + 5.19T + 43T^{2} \) |
| 47 | \( 1 - 3.02T + 47T^{2} \) |
| 53 | \( 1 + 4.00T + 53T^{2} \) |
| 59 | \( 1 + 1.12T + 59T^{2} \) |
| 61 | \( 1 + 0.502T + 61T^{2} \) |
| 67 | \( 1 - 6.95T + 67T^{2} \) |
| 71 | \( 1 + 7.61T + 71T^{2} \) |
| 73 | \( 1 - 2.83T + 73T^{2} \) |
| 79 | \( 1 + 1.13T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 5.33T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−7.925991405486214791179910900833, −7.57683346934189247111884670251, −6.75569906641634612681172383197, −5.29994409572182959457569311459, −5.18076773486197199055844674193, −4.73846246252172073555803970251, −3.38348235019312688155356020534, −2.70821253609760353015099457200, −1.68375437364555422316487971391, −0.47558907843852016090683361844,
0.47558907843852016090683361844, 1.68375437364555422316487971391, 2.70821253609760353015099457200, 3.38348235019312688155356020534, 4.73846246252172073555803970251, 5.18076773486197199055844674193, 5.29994409572182959457569311459, 6.75569906641634612681172383197, 7.57683346934189247111884670251, 7.925991405486214791179910900833