L(s) = 1 | − i·2-s − 4-s − i·7-s + i·8-s + 4·11-s − 6i·13-s − 14-s + 16-s − 2i·17-s + 4·19-s − 4i·22-s + 8i·23-s − 6·26-s + i·28-s − 2·29-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.377i·7-s + 0.353i·8-s + 1.20·11-s − 1.66i·13-s − 0.267·14-s + 0.250·16-s − 0.485i·17-s + 0.917·19-s − 0.852i·22-s + 1.66i·23-s − 1.17·26-s + 0.188i·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.794713349\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.794713349\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−8.575688041178342675376580720894, −7.58708883784009702265886652514, −7.24352896766114055981756508196, −5.88122106670547835414241782326, −5.44936849187633603931885850179, −4.34336573609305764327316516822, −3.54646280687156728472762582249, −2.91013921001833239425327250775, −1.56770016999091544996877878190, −0.65489759606315591206840315201,
1.18213511157640488656663341229, 2.31982376803787028489023508474, 3.65610088890848707296211559053, 4.32895092309883762214038376186, 5.10187447989534779390674500199, 6.18226285667188940045780795352, 6.59696200645346197085099445930, 7.24146641322850673576644518077, 8.310515878353982249337371850513, 8.825662352621021287999239670911