L(s) = 1 | − i·2-s − 4-s − i·7-s + i·8-s − i·13-s − 14-s + 16-s − 3i·17-s − 2·19-s + 3i·23-s − 26-s + i·28-s − 3·29-s − 31-s − i·32-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.377i·7-s + 0.353i·8-s − 0.277i·13-s − 0.267·14-s + 0.250·16-s − 0.727i·17-s − 0.458·19-s + 0.625i·23-s − 0.196·26-s + 0.188i·28-s − 0.557·29-s − 0.179·31-s − 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5489423164\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5489423164\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 3iT - 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 7iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 15iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−8.365065907294969951319741632313, −7.53796247359455911496786008721, −6.88069824419769952242560770924, −5.79772447578655773769943597364, −5.09722438654297425079593523972, −4.17885976969187413867762469923, −3.44089148561824895520842404339, −2.49738694601324945227769951780, −1.45381212515555590144940713589, −0.17003627585707894682212660472,
1.50070957336960335228174065644, 2.68174709079835084957881115169, 3.79512642451380361793118861554, 4.56497589414483073582353861239, 5.40919411310355706303595223420, 6.20666347786653201093634544482, 6.73372141171708547299939532131, 7.67054796206946527669475760269, 8.332642460642684211519171969751, 8.926240425076139529861135002547