| L(s) = 1 | − 2.69·2-s − 1.24·3-s + 5.23·4-s + 2.30·5-s + 3.35·6-s − 0.0663·7-s − 8.71·8-s − 1.44·9-s − 6.21·10-s − 5.57·11-s − 6.53·12-s + 1.28·13-s + 0.178·14-s − 2.87·15-s + 12.9·16-s − 4.04·17-s + 3.89·18-s + 2.99·19-s + 12.0·20-s + 0.0826·21-s + 14.9·22-s + 1.40·23-s + 10.8·24-s + 0.332·25-s − 3.45·26-s + 5.54·27-s − 0.347·28-s + ⋯ |
| L(s) = 1 | − 1.90·2-s − 0.719·3-s + 2.61·4-s + 1.03·5-s + 1.36·6-s − 0.0250·7-s − 3.08·8-s − 0.481·9-s − 1.96·10-s − 1.67·11-s − 1.88·12-s + 0.355·13-s + 0.0476·14-s − 0.743·15-s + 3.24·16-s − 0.982·17-s + 0.916·18-s + 0.686·19-s + 2.70·20-s + 0.0180·21-s + 3.19·22-s + 0.292·23-s + 2.21·24-s + 0.0665·25-s − 0.677·26-s + 1.06·27-s − 0.0656·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2671 | \( 1 + T \) |
| good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 3 | \( 1 + 1.24T + 3T^{2} \) |
| 5 | \( 1 - 2.30T + 5T^{2} \) |
| 7 | \( 1 + 0.0663T + 7T^{2} \) |
| 11 | \( 1 + 5.57T + 11T^{2} \) |
| 13 | \( 1 - 1.28T + 13T^{2} \) |
| 17 | \( 1 + 4.04T + 17T^{2} \) |
| 19 | \( 1 - 2.99T + 19T^{2} \) |
| 23 | \( 1 - 1.40T + 23T^{2} \) |
| 29 | \( 1 - 3.07T + 29T^{2} \) |
| 31 | \( 1 - 8.57T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 7.73T + 41T^{2} \) |
| 43 | \( 1 + 6.31T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 - 4.49T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 7.23T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 - 9.65T + 71T^{2} \) |
| 73 | \( 1 - 3.66T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 + 2.81T + 83T^{2} \) |
| 89 | \( 1 - 4.08T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−8.400940458907905565644982047910, −8.057357271376751108485070557380, −7.00920834706176700631883086599, −6.31309140744400694773983654599, −5.75001070649608844791401603335, −4.91645819323425502177397241041, −2.86209813312262529966227988436, −2.40626273031768490839747513877, −1.15217998038351249377387224182, 0,
1.15217998038351249377387224182, 2.40626273031768490839747513877, 2.86209813312262529966227988436, 4.91645819323425502177397241041, 5.75001070649608844791401603335, 6.31309140744400694773983654599, 7.00920834706176700631883086599, 8.057357271376751108485070557380, 8.400940458907905565644982047910
