| L(s) = 1 | − 1.41·2-s − 1.41·3-s − 1.41·5-s + 2.00·6-s − 7-s + 2.82·8-s − 0.999·9-s + 2.00·10-s + 0.828·11-s + 1.17·13-s + 1.41·14-s + 2.00·15-s − 4.00·16-s + 4.65·17-s + 1.41·18-s − 4.65·19-s + 1.41·21-s − 1.17·22-s − 8.82·23-s − 4·24-s − 2.99·25-s − 1.65·26-s + 5.65·27-s − 1.17·29-s − 2.82·30-s + ⋯ |
| L(s) = 1 | − 1.00·2-s − 0.816·3-s − 0.632·5-s + 0.816·6-s − 0.377·7-s + 0.999·8-s − 0.333·9-s + 0.632·10-s + 0.249·11-s + 0.324·13-s + 0.377·14-s + 0.516·15-s − 1.00·16-s + 1.12·17-s + 0.333·18-s − 1.06·19-s + 0.308·21-s − 0.249·22-s − 1.84·23-s − 0.816·24-s − 0.599·25-s − 0.324·26-s + 1.08·27-s − 0.217·29-s − 0.516·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 4003 | \( 1+O(T) \) |
| good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 0.828T + 11T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 17 | \( 1 - 4.65T + 17T^{2} \) |
| 19 | \( 1 + 4.65T + 19T^{2} \) |
| 23 | \( 1 + 8.82T + 23T^{2} \) |
| 29 | \( 1 + 1.17T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 4.17T + 37T^{2} \) |
| 41 | \( 1 - 5.17T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 3.07T + 47T^{2} \) |
| 53 | \( 1 - 9.31T + 53T^{2} \) |
| 59 | \( 1 + 8.82T + 59T^{2} \) |
| 61 | \( 1 - 7.17T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 3T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 17.1T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 1.75T + 97T^{2} \) |
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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
−8.197238268154793147296299303082, −7.63008765758379215224417701137, −6.55663777766642901951761733324, −6.07297894844572667204275783056, −5.13954118223872036146552147011, −4.23832529337291424040604495150, −3.56605559268132397838855381812, −2.18466034537402874734904992469, −0.898079468955872573049789736613, 0,
0.898079468955872573049789736613, 2.18466034537402874734904992469, 3.56605559268132397838855381812, 4.23832529337291424040604495150, 5.13954118223872036146552147011, 6.07297894844572667204275783056, 6.55663777766642901951761733324, 7.63008765758379215224417701137, 8.197238268154793147296299303082
