L(s) = 1 | + 2-s − 3-s + 4-s + 3.69·5-s − 6-s − 2.24·7-s + 8-s + 9-s + 3.69·10-s − 12-s + 5.56·13-s − 2.24·14-s − 3.69·15-s + 16-s − 1.01·17-s + 18-s − 2.65·19-s + 3.69·20-s + 2.24·21-s + 2.65·23-s − 24-s + 8.62·25-s + 5.56·26-s − 27-s − 2.24·28-s − 1.38·29-s − 3.69·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.65·5-s − 0.408·6-s − 0.847·7-s + 0.353·8-s + 0.333·9-s + 1.16·10-s − 0.288·12-s + 1.54·13-s − 0.599·14-s − 0.953·15-s + 0.250·16-s − 0.246·17-s + 0.235·18-s − 0.608·19-s + 0.825·20-s + 0.489·21-s + 0.552·23-s − 0.204·24-s + 1.72·25-s + 1.09·26-s − 0.192·27-s − 0.423·28-s − 0.256·29-s − 0.673·30-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
\(L(1)\) |
\(\approx\) |
\(2.666722477\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.666722477\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.69T + 5T^{2} \) |
| 7 | \( 1 + 2.24T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 5.56T + 13T^{2} \) |
| 17 | \( 1 + 1.01T + 17T^{2} \) |
| 19 | \( 1 + 2.65T + 19T^{2} \) |
| 23 | \( 1 - 2.65T + 23T^{2} \) |
| 29 | \( 1 + 1.38T + 29T^{2} \) |
| 31 | \( 1 - 4.89T + 31T^{2} \) |
| 37 | \( 1 + 1.97T + 37T^{2} \) |
| 41 | \( 1 - 4.28T + 41T^{2} \) |
| 43 | \( 1 - 0.551T + 43T^{2} \) |
| 47 | \( 1 + 5.62T + 47T^{2} \) |
| 53 | \( 1 - 1.03T + 53T^{2} \) |
| 59 | \( 1 + 2.42T + 59T^{2} \) |
| 61 | \( 1 - 5.38T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 + 8.68T + 71T^{2} \) |
| 73 | \( 1 + 1.75T + 73T^{2} \) |
| 79 | \( 1 - 3.74T + 79T^{2} \) |
| 83 | \( 1 + 5.43T + 83T^{2} \) |
| 89 | \( 1 + 6.27T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−10.09348347700654059110841597279, −9.347798045776330607239148557166, −8.436971867988287508527760455222, −6.89159768123836562322937457695, −6.29014790976745765921335573631, −5.86013421130063564835559368694, −4.92477451535507171290009692315, −3.72241063904949797193709941326, −2.57001594144827196369347354630, −1.35272616330164946289392516319,
1.35272616330164946289392516319, 2.57001594144827196369347354630, 3.72241063904949797193709941326, 4.92477451535507171290009692315, 5.86013421130063564835559368694, 6.29014790976745765921335573631, 6.89159768123836562322937457695, 8.436971867988287508527760455222, 9.347798045776330607239148557166, 10.09348347700654059110841597279