| L(s) = 1 | + 2-s + 0.702·3-s + 4-s + 2.22·5-s + 0.702·6-s − 1.98·7-s + 8-s − 2.50·9-s + 2.22·10-s − 4.78·11-s + 0.702·12-s − 0.605·13-s − 1.98·14-s + 1.56·15-s + 16-s + 4.18·17-s − 2.50·18-s + 6.41·19-s + 2.22·20-s − 1.39·21-s − 4.78·22-s − 6.67·23-s + 0.702·24-s − 0.0576·25-s − 0.605·26-s − 3.86·27-s − 1.98·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.405·3-s + 0.5·4-s + 0.994·5-s + 0.286·6-s − 0.751·7-s + 0.353·8-s − 0.835·9-s + 0.703·10-s − 1.44·11-s + 0.202·12-s − 0.167·13-s − 0.531·14-s + 0.403·15-s + 0.250·16-s + 1.01·17-s − 0.590·18-s + 1.47·19-s + 0.497·20-s − 0.305·21-s − 1.02·22-s − 1.39·23-s + 0.143·24-s − 0.0115·25-s − 0.118·26-s − 0.744·27-s − 0.375·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3001 | \( 1+O(T) \) |
| good | 3 | \( 1 - 0.702T + 3T^{2} \) |
| 5 | \( 1 - 2.22T + 5T^{2} \) |
| 7 | \( 1 + 1.98T + 7T^{2} \) |
| 11 | \( 1 + 4.78T + 11T^{2} \) |
| 13 | \( 1 + 0.605T + 13T^{2} \) |
| 17 | \( 1 - 4.18T + 17T^{2} \) |
| 19 | \( 1 - 6.41T + 19T^{2} \) |
| 23 | \( 1 + 6.67T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 + 4.67T + 31T^{2} \) |
| 37 | \( 1 - 1.57T + 37T^{2} \) |
| 41 | \( 1 - 1.45T + 41T^{2} \) |
| 43 | \( 1 + 5.97T + 43T^{2} \) |
| 47 | \( 1 - 3.98T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 6.95T + 61T^{2} \) |
| 67 | \( 1 + 2.59T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 + 9.45T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 - 7.11T + 97T^{2} \) |
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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
−7.71227758037427812852642693000, −7.02437740615874407260962241045, −5.82537217249084535668415592870, −5.70285781056509126196272438505, −5.16689487516294000080992647925, −3.79988733602157199915727696094, −3.18887578540024669274949073667, −2.51895420385446292093224067343, −1.73445457027230370699016192281, 0,
1.73445457027230370699016192281, 2.51895420385446292093224067343, 3.18887578540024669274949073667, 3.79988733602157199915727696094, 5.16689487516294000080992647925, 5.70285781056509126196272438505, 5.82537217249084535668415592870, 7.02437740615874407260962241045, 7.71227758037427812852642693000
