| L(s) = 1 | − 2.69·2-s − 3-s + 5.24·4-s + 3.62·5-s + 2.69·6-s − 8.73·8-s + 9-s − 9.76·10-s + 3.79·11-s − 5.24·12-s − 0.986·13-s − 3.62·15-s + 13.0·16-s − 1.05·17-s − 2.69·18-s + 2.81·19-s + 19.0·20-s − 10.2·22-s − 5.66·23-s + 8.73·24-s + 8.16·25-s + 2.65·26-s − 27-s − 2.21·29-s + 9.76·30-s − 10.4·31-s − 17.5·32-s + ⋯ |
| L(s) = 1 | − 1.90·2-s − 0.577·3-s + 2.62·4-s + 1.62·5-s + 1.09·6-s − 3.08·8-s + 0.333·9-s − 3.08·10-s + 1.14·11-s − 1.51·12-s − 0.273·13-s − 0.936·15-s + 3.25·16-s − 0.256·17-s − 0.634·18-s + 0.644·19-s + 4.25·20-s − 2.17·22-s − 1.18·23-s + 1.78·24-s + 1.63·25-s + 0.521·26-s − 0.192·27-s − 0.411·29-s + 1.78·30-s − 1.87·31-s − 3.10·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 5 | \( 1 - 3.62T + 5T^{2} \) |
| 11 | \( 1 - 3.79T + 11T^{2} \) |
| 13 | \( 1 + 0.986T + 13T^{2} \) |
| 17 | \( 1 + 1.05T + 17T^{2} \) |
| 19 | \( 1 - 2.81T + 19T^{2} \) |
| 23 | \( 1 + 5.66T + 23T^{2} \) |
| 29 | \( 1 + 2.21T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 7.86T + 37T^{2} \) |
| 43 | \( 1 - 4.16T + 43T^{2} \) |
| 47 | \( 1 - 5.08T + 47T^{2} \) |
| 53 | \( 1 + 9.17T + 53T^{2} \) |
| 59 | \( 1 - 7.58T + 59T^{2} \) |
| 61 | \( 1 + 5.96T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 3.75T + 73T^{2} \) |
| 79 | \( 1 + 4.83T + 79T^{2} \) |
| 83 | \( 1 + 7.04T + 83T^{2} \) |
| 89 | \( 1 + 9.53T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−7.73513104358092857788240839561, −7.02522546683786097452227378674, −6.53270003871084495530750576473, −5.83628038338227858471031059145, −5.36192460779034412495817545325, −3.86946000680304715194578836025, −2.66219813059146547267890607767, −1.73627937499180000444989750799, −1.40865716418426768386447537870, 0,
1.40865716418426768386447537870, 1.73627937499180000444989750799, 2.66219813059146547267890607767, 3.86946000680304715194578836025, 5.36192460779034412495817545325, 5.83628038338227858471031059145, 6.53270003871084495530750576473, 7.02522546683786097452227378674, 7.73513104358092857788240839561
