L(s) = 1 | + (−0.366 − 0.366i)2-s + (−0.5 + 0.866i)3-s − 0.732i·4-s + (0.5 − 0.133i)6-s + (−0.633 + 0.633i)8-s + (−0.499 − 0.866i)9-s + (0.633 + 0.366i)12-s + i·13-s − 0.267·16-s + (−0.133 + 0.499i)18-s + i·23-s + (−0.232 − 0.866i)24-s + 25-s + (0.366 − 0.366i)26-s + 0.999·27-s + ⋯ |
L(s) = 1 | + (−0.366 − 0.366i)2-s + (−0.5 + 0.866i)3-s − 0.732i·4-s + (0.5 − 0.133i)6-s + (−0.633 + 0.633i)8-s + (−0.499 − 0.866i)9-s + (0.633 + 0.366i)12-s + i·13-s − 0.267·16-s + (−0.133 + 0.499i)18-s + i·23-s + (−0.232 − 0.866i)24-s + 25-s + (0.366 − 0.366i)26-s + 0.999·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7381606809\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7381606809\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 + (-0.866 - 0.5i)T \) |
good | 2 | \( 1 + (0.366 + 0.366i)T + iT^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 13 | \( 1 - iT - T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 31 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (1.36 - 1.36i)T - iT^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + 2iT - T^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - 1.73T + T^{2} \) |
| 73 | \( 1 + (-1.36 - 1.36i)T + iT^{2} \) |
| 79 | \( 1 + iT^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.635281251366485370352921435145, −8.904363434354504577400248109405, −8.128911597441672183842904856222, −6.67418707408202703403184723180, −6.29830716367017041866921438655, −5.18410203319351804952182956584, −4.71607887958289950662908195534, −3.60951970530572468363066648295, −2.47051027480939710241607591930, −1.10676158517570386946840708730,
0.795805904923888616144521687152, 2.45778437444825477756707638621, 3.24421403258479044037726120283, 4.57906979751940263258438758817, 5.45312103064552829472439465276, 6.53535344651417006818297298832, 6.86100795317575632897686229792, 7.79447343708667465739090896143, 8.399312595516390809794310351878, 8.877697691907082827489808676285