| L(s) = 1 | + (0.366 − 1.36i)2-s + (−1.73 − i)4-s − 0.732i·7-s + (−2 + 1.99i)8-s + 2i·11-s + 3.46·13-s + (−1 − 0.267i)14-s + (1.99 + 3.46i)16-s + 3.46i·17-s − 0.535i·19-s + (2.73 + 0.732i)22-s + 6.19i·23-s + (1.26 − 4.73i)26-s + (−0.732 + 1.26i)28-s + 6.92i·29-s + ⋯ |
| L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.866 − 0.5i)4-s − 0.276i·7-s + (−0.707 + 0.707i)8-s + 0.603i·11-s + 0.960·13-s + (−0.267 − 0.0716i)14-s + (0.499 + 0.866i)16-s + 0.840i·17-s − 0.122i·19-s + (0.582 + 0.156i)22-s + 1.29i·23-s + (0.248 − 0.928i)26-s + (−0.138 + 0.239i)28-s + 1.28i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.611202255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.611202255\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.366 + 1.36i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 0.732iT - 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 0.535iT - 19T^{2} \) |
| 23 | \( 1 - 6.19iT - 23T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 1.46T + 41T^{2} \) |
| 43 | \( 1 - 5.26T + 43T^{2} \) |
| 47 | \( 1 - 3.26iT - 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + 7.46iT - 59T^{2} \) |
| 61 | \( 1 + 8.92iT - 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 5.46T + 71T^{2} \) |
| 73 | \( 1 - 7.46iT - 73T^{2} \) |
| 79 | \( 1 - 1.07T + 79T^{2} \) |
| 83 | \( 1 - 1.26T + 83T^{2} \) |
| 89 | \( 1 - 8.92T + 89T^{2} \) |
| 97 | \( 1 - 14.3iT - 97T^{2} \) |
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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
−9.302868907941067423038489179328, −8.720311083455176974554734493055, −7.78107294785437780506178668068, −6.80132734858067526401386003109, −5.77019656136358893423851221171, −5.07421026850103707576951762632, −3.93661535712193837689058296910, −3.47856842984895355613319264577, −2.11488947378262746748655621640, −1.18688230191647644284742036211,
0.64054266283646161527207796357, 2.54325279081325834277249328822, 3.65479901934812892995093085235, 4.43065472863343550565325607578, 5.52252923029711317383415680308, 6.01679419007930889797554109180, 6.90293695825753977546108439908, 7.65925186894731476926242097225, 8.644087300486921044357861895415, 8.868677419943318162742661608166
