| L(s) = 1 | − 5.35·2-s − 3·3-s + 20.6·4-s + 11.7·5-s + 16.0·6-s + 34.3·7-s − 67.7·8-s + 9·9-s − 63.1·10-s − 22.6·11-s − 61.9·12-s − 45.8·13-s − 183.·14-s − 35.3·15-s + 197.·16-s − 59.5·17-s − 48.1·18-s + 19.1·19-s + 243.·20-s − 102.·21-s + 121.·22-s + 23·23-s + 203.·24-s + 14.1·25-s + 245.·26-s − 27·27-s + 709.·28-s + ⋯ |
| L(s) = 1 | − 1.89·2-s − 0.577·3-s + 2.58·4-s + 1.05·5-s + 1.09·6-s + 1.85·7-s − 2.99·8-s + 0.333·9-s − 1.99·10-s − 0.622·11-s − 1.49·12-s − 0.977·13-s − 3.50·14-s − 0.609·15-s + 3.08·16-s − 0.849·17-s − 0.630·18-s + 0.231·19-s + 2.72·20-s − 1.07·21-s + 1.17·22-s + 0.208·23-s + 1.72·24-s + 0.112·25-s + 1.85·26-s − 0.192·27-s + 4.78·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 23 | \( 1 - 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 + 5.35T + 8T^{2} \) |
| 5 | \( 1 - 11.7T + 125T^{2} \) |
| 7 | \( 1 - 34.3T + 343T^{2} \) |
| 11 | \( 1 + 22.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 45.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 59.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 19.1T + 6.85e3T^{2} \) |
| 31 | \( 1 - 65.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 24.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 129.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 268.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 222.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 124.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 581.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 569.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 425.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 502.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 282.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 88.1T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.01e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.61e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 772.T + 9.12e5T^{2} \) |
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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
−8.436255976819127655110659775754, −7.75651835169793041661418958245, −7.18858077833649896766422888982, −6.23540781279979053379893690329, −5.40581621847049052744363569618, −4.65172991612612636785461359571, −2.58501507110703584363164800239, −1.94003669230739234483074318554, −1.22011380880928109702789378770, 0,
1.22011380880928109702789378770, 1.94003669230739234483074318554, 2.58501507110703584363164800239, 4.65172991612612636785461359571, 5.40581621847049052744363569618, 6.23540781279979053379893690329, 7.18858077833649896766422888982, 7.75651835169793041661418958245, 8.436255976819127655110659775754
