| L(s) = 1 | + 0.232·2-s + 2.39·3-s − 1.94·4-s + 0.556·6-s + 2.06·7-s − 0.917·8-s + 2.73·9-s + 0.480·11-s − 4.65·12-s − 4.07·13-s + 0.480·14-s + 3.67·16-s + 0.635·18-s + 4·19-s + 4.94·21-s + 0.111·22-s + 8.15·23-s − 2.19·24-s − 0.945·26-s − 0.639·27-s − 4.01·28-s − 1.03·29-s + 6.06·31-s + 2.68·32-s + 1.14·33-s − 5.31·36-s − 1.29·37-s + ⋯ |
| L(s) = 1 | + 0.164·2-s + 1.38·3-s − 0.972·4-s + 0.227·6-s + 0.780·7-s − 0.324·8-s + 0.910·9-s + 0.144·11-s − 1.34·12-s − 1.12·13-s + 0.128·14-s + 0.919·16-s + 0.149·18-s + 0.917·19-s + 1.07·21-s + 0.0237·22-s + 1.69·23-s − 0.448·24-s − 0.185·26-s − 0.123·27-s − 0.759·28-s − 0.192·29-s + 1.08·31-s + 0.475·32-s + 0.200·33-s − 0.886·36-s − 0.213·37-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
| \(L(1)\) |
\(\approx\) |
\(3.211390719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.211390719\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.232T + 2T^{2} \) |
| 3 | \( 1 - 2.39T + 3T^{2} \) |
| 7 | \( 1 - 2.06T + 7T^{2} \) |
| 11 | \( 1 - 0.480T + 11T^{2} \) |
| 13 | \( 1 + 4.07T + 13T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 8.15T + 23T^{2} \) |
| 29 | \( 1 + 1.03T + 29T^{2} \) |
| 31 | \( 1 - 6.06T + 31T^{2} \) |
| 37 | \( 1 + 1.29T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 7.45T + 43T^{2} \) |
| 47 | \( 1 + 3.60T + 47T^{2} \) |
| 53 | \( 1 + 6.14T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 + 3.14T + 67T^{2} \) |
| 71 | \( 1 + 1.81T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 2.23T + 83T^{2} \) |
| 89 | \( 1 + 9.37T + 89T^{2} \) |
| 97 | \( 1 - 16.7T + 97T^{2} \) |
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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
−7.934503522319347058141382895254, −7.58871691548618659316014456427, −6.71564636289427509462766191179, −5.54931047453499379409745463969, −4.84941744689678934009567936112, −4.44677078857767205843993513076, −3.36396555676508735829953860364, −2.94880976533262965267614925341, −1.93571347396201792911691822694, −0.855284509943932983493051417777,
0.855284509943932983493051417777, 1.93571347396201792911691822694, 2.94880976533262965267614925341, 3.36396555676508735829953860364, 4.44677078857767205843993513076, 4.84941744689678934009567936112, 5.54931047453499379409745463969, 6.71564636289427509462766191179, 7.58871691548618659316014456427, 7.934503522319347058141382895254
