| L(s) = 1 | − 2.47·2-s + 2.84·3-s + 4.14·4-s − 3.76·5-s − 7.04·6-s − 7-s − 5.31·8-s + 5.06·9-s + 9.32·10-s + 3.19·11-s + 11.7·12-s + 6.78·13-s + 2.47·14-s − 10.6·15-s + 4.89·16-s + 0.305·17-s − 12.5·18-s + 2.42·19-s − 15.5·20-s − 2.84·21-s − 7.90·22-s + 6.84·23-s − 15.1·24-s + 9.13·25-s − 16.8·26-s + 5.86·27-s − 4.14·28-s + ⋯ |
| L(s) = 1 | − 1.75·2-s + 1.63·3-s + 2.07·4-s − 1.68·5-s − 2.87·6-s − 0.377·7-s − 1.88·8-s + 1.68·9-s + 2.94·10-s + 0.961·11-s + 3.39·12-s + 1.88·13-s + 0.662·14-s − 2.75·15-s + 1.22·16-s + 0.0741·17-s − 2.96·18-s + 0.556·19-s − 3.48·20-s − 0.619·21-s − 1.68·22-s + 1.42·23-s − 3.08·24-s + 1.82·25-s − 3.29·26-s + 1.12·27-s − 0.783·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8354890828\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8354890828\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 2.47T + 2T^{2} \) |
| 3 | \( 1 - 2.84T + 3T^{2} \) |
| 5 | \( 1 + 3.76T + 5T^{2} \) |
| 11 | \( 1 - 3.19T + 11T^{2} \) |
| 13 | \( 1 - 6.78T + 13T^{2} \) |
| 17 | \( 1 - 0.305T + 17T^{2} \) |
| 19 | \( 1 - 2.42T + 19T^{2} \) |
| 23 | \( 1 - 6.84T + 23T^{2} \) |
| 29 | \( 1 + 1.18T + 29T^{2} \) |
| 31 | \( 1 + 6.37T + 31T^{2} \) |
| 37 | \( 1 + 3.49T + 37T^{2} \) |
| 43 | \( 1 - 2.03T + 43T^{2} \) |
| 47 | \( 1 - 0.930T + 47T^{2} \) |
| 53 | \( 1 + 5.48T + 53T^{2} \) |
| 59 | \( 1 + 1.01T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 - 7.91T + 67T^{2} \) |
| 71 | \( 1 + 1.64T + 71T^{2} \) |
| 73 | \( 1 - 6.23T + 73T^{2} \) |
| 79 | \( 1 - 6.61T + 79T^{2} \) |
| 83 | \( 1 - 7.93T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 + 9.52T + 97T^{2} \) |
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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
−11.36635241153884565967464434850, −10.79629687299796241925014334204, −9.319450235100998489163066137823, −8.933747238189787890539431619566, −8.214230780377831061652897103468, −7.49514859744915059707363240429, −6.69849624576113448195592076551, −3.86153618672425635165382686946, −3.17653546942301702643588902945, −1.27871960126229417602407470457,
1.27871960126229417602407470457, 3.17653546942301702643588902945, 3.86153618672425635165382686946, 6.69849624576113448195592076551, 7.49514859744915059707363240429, 8.214230780377831061652897103468, 8.933747238189787890539431619566, 9.319450235100998489163066137823, 10.79629687299796241925014334204, 11.36635241153884565967464434850
