| L(s) = 1 | − 2.58·2-s + 2.93·3-s + 4.68·4-s + 1.26·5-s − 7.60·6-s + 7-s − 6.94·8-s + 5.64·9-s − 3.25·10-s + 5.67·11-s + 13.7·12-s − 2.58·14-s + 3.70·15-s + 8.58·16-s − 1.07·17-s − 14.5·18-s − 0.612·19-s + 5.90·20-s + 2.93·21-s − 14.6·22-s + 3.02·23-s − 20.4·24-s − 3.41·25-s + 7.77·27-s + 4.68·28-s + 1.64·29-s − 9.58·30-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 1.69·3-s + 2.34·4-s + 0.563·5-s − 3.10·6-s + 0.377·7-s − 2.45·8-s + 1.88·9-s − 1.03·10-s + 1.71·11-s + 3.97·12-s − 0.691·14-s + 0.956·15-s + 2.14·16-s − 0.260·17-s − 3.43·18-s − 0.140·19-s + 1.32·20-s + 0.641·21-s − 3.12·22-s + 0.631·23-s − 4.16·24-s − 0.682·25-s + 1.49·27-s + 0.885·28-s + 0.304·29-s − 1.74·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.673735562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.673735562\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.58T + 2T^{2} \) |
| 3 | \( 1 - 2.93T + 3T^{2} \) |
| 5 | \( 1 - 1.26T + 5T^{2} \) |
| 11 | \( 1 - 5.67T + 11T^{2} \) |
| 17 | \( 1 + 1.07T + 17T^{2} \) |
| 19 | \( 1 + 0.612T + 19T^{2} \) |
| 23 | \( 1 - 3.02T + 23T^{2} \) |
| 29 | \( 1 - 1.64T + 29T^{2} \) |
| 31 | \( 1 + 8.21T + 31T^{2} \) |
| 37 | \( 1 - 4.81T + 37T^{2} \) |
| 41 | \( 1 - 0.993T + 41T^{2} \) |
| 43 | \( 1 + 4.96T + 43T^{2} \) |
| 47 | \( 1 - 3.93T + 47T^{2} \) |
| 53 | \( 1 - 3.04T + 53T^{2} \) |
| 59 | \( 1 + 7.61T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 0.519T + 71T^{2} \) |
| 73 | \( 1 + 15.9T + 73T^{2} \) |
| 79 | \( 1 + 4.51T + 79T^{2} \) |
| 83 | \( 1 + 3.75T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 - 7.28T + 97T^{2} \) |
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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
−9.404793818683122807571622908616, −9.007736725100766758356328207306, −8.474165183902347973552542522580, −7.57388744510826409134562358170, −6.99413840099479390649269667283, −6.05660956033723399652769983515, −4.19708028744456486840756907858, −3.07883385743895788835289926214, −1.99957560430501647550192946452, −1.38586111002966411124216874537,
1.38586111002966411124216874537, 1.99957560430501647550192946452, 3.07883385743895788835289926214, 4.19708028744456486840756907858, 6.05660956033723399652769983515, 6.99413840099479390649269667283, 7.57388744510826409134562358170, 8.474165183902347973552542522580, 9.007736725100766758356328207306, 9.404793818683122807571622908616
