| L(s) = 1 | − 1.90·2-s − 1.08·3-s + 1.63·4-s + 1.10·5-s + 2.06·6-s − 7-s + 0.687·8-s − 1.82·9-s − 2.10·10-s + 0.799·11-s − 1.77·12-s + 1.90·14-s − 1.19·15-s − 4.59·16-s + 2.63·17-s + 3.47·18-s + 1.84·19-s + 1.81·20-s + 1.08·21-s − 1.52·22-s − 3.82·23-s − 0.745·24-s − 3.77·25-s + 5.23·27-s − 1.63·28-s − 7.54·29-s + 2.28·30-s + ⋯ |
| L(s) = 1 | − 1.34·2-s − 0.626·3-s + 0.819·4-s + 0.494·5-s + 0.845·6-s − 0.377·7-s + 0.242·8-s − 0.607·9-s − 0.667·10-s + 0.241·11-s − 0.513·12-s + 0.509·14-s − 0.309·15-s − 1.14·16-s + 0.640·17-s + 0.819·18-s + 0.422·19-s + 0.405·20-s + 0.236·21-s − 0.325·22-s − 0.797·23-s − 0.152·24-s − 0.755·25-s + 1.00·27-s − 0.309·28-s − 1.40·29-s + 0.417·30-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.90T + 2T^{2} \) |
| 3 | \( 1 + 1.08T + 3T^{2} \) |
| 5 | \( 1 - 1.10T + 5T^{2} \) |
| 11 | \( 1 - 0.799T + 11T^{2} \) |
| 17 | \( 1 - 2.63T + 17T^{2} \) |
| 19 | \( 1 - 1.84T + 19T^{2} \) |
| 23 | \( 1 + 3.82T + 23T^{2} \) |
| 29 | \( 1 + 7.54T + 29T^{2} \) |
| 31 | \( 1 - 7.86T + 31T^{2} \) |
| 37 | \( 1 + 3.45T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 2.63T + 43T^{2} \) |
| 47 | \( 1 + 5.22T + 47T^{2} \) |
| 53 | \( 1 - 6.15T + 53T^{2} \) |
| 59 | \( 1 + 5.10T + 59T^{2} \) |
| 61 | \( 1 - 3.31T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 + 6.69T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 + 8.10T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 - 2.27T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−9.566276799589942650406161107366, −8.614056484554483587248967438639, −7.898508966014172754866398714165, −7.00290902508849158501389440951, −6.06458845820133152635519741187, −5.42063910847353780179631120426, −4.09550311191270568010283564646, −2.66237447124878453066153415998, −1.36366338077460749421270028786, 0,
1.36366338077460749421270028786, 2.66237447124878453066153415998, 4.09550311191270568010283564646, 5.42063910847353780179631120426, 6.06458845820133152635519741187, 7.00290902508849158501389440951, 7.898508966014172754866398714165, 8.614056484554483587248967438639, 9.566276799589942650406161107366
