L(s) = 1 | + 1.73i·5-s + 1.73i·7-s − 3·11-s − 5·13-s − 6.92i·17-s + 3.46i·19-s − 9·23-s + 2.00·25-s − 1.73i·29-s − 5.19i·31-s − 2.99·35-s + 2·37-s − 5.19i·41-s − 5.19i·43-s − 3·47-s + ⋯ |
L(s) = 1 | + 0.774i·5-s + 0.654i·7-s − 0.904·11-s − 1.38·13-s − 1.68i·17-s + 0.794i·19-s − 1.87·23-s + 0.400·25-s − 0.321i·29-s − 0.933i·31-s − 0.507·35-s + 0.328·37-s − 0.811i·41-s − 0.792i·43-s − 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 7 | \( 1 - 1.73iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 + 6.92iT - 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 9T + 23T^{2} \) |
| 29 | \( 1 + 1.73iT - 29T^{2} \) |
| 31 | \( 1 + 5.19iT - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 5.19iT - 41T^{2} \) |
| 43 | \( 1 + 5.19iT - 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - 8.66iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 8.66iT - 79T^{2} \) |
| 83 | \( 1 + 15T + 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 + 5T + 97T^{2} \) |
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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
−9.586951606307408230635399731894, −8.408440757382028679925142963636, −7.57849730887031902866492749239, −7.03033742323893355357463373614, −5.85825779106772864640362790505, −5.23467684161647150424458647281, −4.12316805432960487693612068676, −2.74806353825111965562245834556, −2.31373520905806898313826459612, 0,
1.60458615116037850498349215531, 2.85008212785772591111458821328, 4.18000835834928814000000770599, 4.81346418523860894126606052481, 5.73924896975093824779277616479, 6.75941712399394419586274041128, 7.72999922642322778788813667864, 8.244826003770117761261607900327, 9.177737645232944109451932673915