| L(s) = 1 | − 2.27·2-s − 3.13·3-s + 3.16·4-s + 7.12·6-s + 4.34·7-s − 2.64·8-s + 6.83·9-s − 3.85·11-s − 9.92·12-s + 3.05·13-s − 9.88·14-s − 0.321·16-s − 3.04·17-s − 15.5·18-s + 2.69·19-s − 13.6·21-s + 8.74·22-s − 23-s + 8.28·24-s − 6.94·26-s − 12.0·27-s + 13.7·28-s + 3.86·29-s − 4.75·31-s + 6.01·32-s + 12.0·33-s + 6.91·34-s + ⋯ |
| L(s) = 1 | − 1.60·2-s − 1.81·3-s + 1.58·4-s + 2.90·6-s + 1.64·7-s − 0.934·8-s + 2.27·9-s − 1.16·11-s − 2.86·12-s + 0.847·13-s − 2.64·14-s − 0.0804·16-s − 0.738·17-s − 3.66·18-s + 0.618·19-s − 2.97·21-s + 1.86·22-s − 0.208·23-s + 1.69·24-s − 1.36·26-s − 2.31·27-s + 2.59·28-s + 0.717·29-s − 0.854·31-s + 1.06·32-s + 2.10·33-s + 1.18·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3983050916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3983050916\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 2.27T + 2T^{2} \) |
| 3 | \( 1 + 3.13T + 3T^{2} \) |
| 7 | \( 1 - 4.34T + 7T^{2} \) |
| 11 | \( 1 + 3.85T + 11T^{2} \) |
| 13 | \( 1 - 3.05T + 13T^{2} \) |
| 17 | \( 1 + 3.04T + 17T^{2} \) |
| 19 | \( 1 - 2.69T + 19T^{2} \) |
| 29 | \( 1 - 3.86T + 29T^{2} \) |
| 31 | \( 1 + 4.75T + 31T^{2} \) |
| 37 | \( 1 - 1.06T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 8.40T + 43T^{2} \) |
| 47 | \( 1 - 7.89T + 47T^{2} \) |
| 53 | \( 1 + 2.82T + 53T^{2} \) |
| 59 | \( 1 + 7.06T + 59T^{2} \) |
| 61 | \( 1 + 4.81T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 7.89T + 71T^{2} \) |
| 73 | \( 1 + 4.83T + 73T^{2} \) |
| 79 | \( 1 - 4.94T + 79T^{2} \) |
| 83 | \( 1 + 2.56T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 - 6.68T + 97T^{2} \) |
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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
−10.83356516446245166072847479533, −10.16210930826735161028697116445, −8.984254062161819996514426505139, −7.975203129932502542605002635481, −7.40976892016690366680403839482, −6.31510343295236119747661547065, −5.31922968043146306258128249299, −4.51817262682821760601975464110, −1.92893206128610051484883692764, −0.798760824816798734139299422139,
0.798760824816798734139299422139, 1.92893206128610051484883692764, 4.51817262682821760601975464110, 5.31922968043146306258128249299, 6.31510343295236119747661547065, 7.40976892016690366680403839482, 7.975203129932502542605002635481, 8.984254062161819996514426505139, 10.16210930826735161028697116445, 10.83356516446245166072847479533
