| L(s) = 1 | + 2-s + 3-s + 4-s + 3.29·5-s + 6-s − 1.34·7-s + 8-s + 9-s + 3.29·10-s + 5.12·11-s + 12-s − 1.72·13-s − 1.34·14-s + 3.29·15-s + 16-s − 0.817·17-s + 18-s − 8.07·19-s + 3.29·20-s − 1.34·21-s + 5.12·22-s − 4.72·23-s + 24-s + 5.86·25-s − 1.72·26-s + 27-s − 1.34·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.47·5-s + 0.408·6-s − 0.509·7-s + 0.353·8-s + 0.333·9-s + 1.04·10-s + 1.54·11-s + 0.288·12-s − 0.478·13-s − 0.360·14-s + 0.851·15-s + 0.250·16-s − 0.198·17-s + 0.235·18-s − 1.85·19-s + 0.737·20-s − 0.294·21-s + 1.09·22-s − 0.986·23-s + 0.204·24-s + 1.17·25-s − 0.338·26-s + 0.192·27-s − 0.254·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.685413245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.685413245\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 3.29T + 5T^{2} \) |
| 7 | \( 1 + 1.34T + 7T^{2} \) |
| 11 | \( 1 - 5.12T + 11T^{2} \) |
| 13 | \( 1 + 1.72T + 13T^{2} \) |
| 17 | \( 1 + 0.817T + 17T^{2} \) |
| 19 | \( 1 + 8.07T + 19T^{2} \) |
| 23 | \( 1 + 4.72T + 23T^{2} \) |
| 29 | \( 1 - 7.99T + 29T^{2} \) |
| 31 | \( 1 + 6.07T + 31T^{2} \) |
| 37 | \( 1 + 6.06T + 37T^{2} \) |
| 41 | \( 1 + 4.77T + 41T^{2} \) |
| 43 | \( 1 - 5.73T + 43T^{2} \) |
| 47 | \( 1 - 9.77T + 47T^{2} \) |
| 53 | \( 1 + 1.53T + 53T^{2} \) |
| 59 | \( 1 - 3.30T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 7.41T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 5.92T + 83T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 97 | \( 1 - 1.05T + 97T^{2} \) |
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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
−10.03829319368727794278577349628, −9.133538820770319350060754748354, −8.576860523835043581022158805522, −7.09124365894330914440744266510, −6.42601056761757824217222733706, −5.86003590082879404203416422625, −4.58592563776405242781209721865, −3.72947599031362669301071301671, −2.46970769656268088032300682567, −1.70755924716008150835227104162,
1.70755924716008150835227104162, 2.46970769656268088032300682567, 3.72947599031362669301071301671, 4.58592563776405242781209721865, 5.86003590082879404203416422625, 6.42601056761757824217222733706, 7.09124365894330914440744266510, 8.576860523835043581022158805522, 9.133538820770319350060754748354, 10.03829319368727794278577349628
