| L(s) = 1 | − 2.36·2-s + 0.649·3-s + 3.58·4-s − 0.526·5-s − 1.53·6-s + 1.65·7-s − 3.74·8-s − 2.57·9-s + 1.24·10-s − 0.633·11-s + 2.32·12-s − 4.08·13-s − 3.90·14-s − 0.341·15-s + 1.68·16-s − 1.27·17-s + 6.09·18-s − 0.310·19-s − 1.88·20-s + 1.07·21-s + 1.49·22-s − 3.61·23-s − 2.43·24-s − 4.72·25-s + 9.64·26-s − 3.62·27-s + 5.92·28-s + ⋯ |
| L(s) = 1 | − 1.67·2-s + 0.374·3-s + 1.79·4-s − 0.235·5-s − 0.626·6-s + 0.624·7-s − 1.32·8-s − 0.859·9-s + 0.393·10-s − 0.190·11-s + 0.672·12-s − 1.13·13-s − 1.04·14-s − 0.0882·15-s + 0.421·16-s − 0.309·17-s + 1.43·18-s − 0.0713·19-s − 0.422·20-s + 0.234·21-s + 0.318·22-s − 0.753·23-s − 0.496·24-s − 0.944·25-s + 1.89·26-s − 0.696·27-s + 1.11·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5171764195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5171764195\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4027 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.36T + 2T^{2} \) |
| 3 | \( 1 - 0.649T + 3T^{2} \) |
| 5 | \( 1 + 0.526T + 5T^{2} \) |
| 7 | \( 1 - 1.65T + 7T^{2} \) |
| 11 | \( 1 + 0.633T + 11T^{2} \) |
| 13 | \( 1 + 4.08T + 13T^{2} \) |
| 17 | \( 1 + 1.27T + 17T^{2} \) |
| 19 | \( 1 + 0.310T + 19T^{2} \) |
| 23 | \( 1 + 3.61T + 23T^{2} \) |
| 29 | \( 1 + 2.51T + 29T^{2} \) |
| 31 | \( 1 + 1.97T + 31T^{2} \) |
| 37 | \( 1 - 4.86T + 37T^{2} \) |
| 41 | \( 1 - 2.89T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 8.11T + 47T^{2} \) |
| 53 | \( 1 - 5.26T + 53T^{2} \) |
| 59 | \( 1 - 5.54T + 59T^{2} \) |
| 61 | \( 1 - 4.30T + 61T^{2} \) |
| 67 | \( 1 + 3.11T + 67T^{2} \) |
| 71 | \( 1 - 6.57T + 71T^{2} \) |
| 73 | \( 1 - 0.973T + 73T^{2} \) |
| 79 | \( 1 - 4.86T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−8.409403856261221573845843063551, −7.85110641876829732287459936535, −7.48659415551423123101658244910, −6.55079809057219132121763827646, −5.64665662109602617974506868218, −4.72064839536222707708723013100, −3.60276102446412278246185110599, −2.39274660822808119695550589761, −1.98550491205191257384524952612, −0.49263763499563270221383449184,
0.49263763499563270221383449184, 1.98550491205191257384524952612, 2.39274660822808119695550589761, 3.60276102446412278246185110599, 4.72064839536222707708723013100, 5.64665662109602617974506868218, 6.55079809057219132121763827646, 7.48659415551423123101658244910, 7.85110641876829732287459936535, 8.409403856261221573845843063551
