L(s) = 1 | + (−0.866 − 0.5i)2-s + i·8-s + (0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)16-s + (−0.866 + 0.499i)18-s − 0.999i·22-s + (0.866 + 0.5i)23-s + 29-s + (−0.866 − 0.5i)37-s − i·43-s + (−0.499 − 0.866i)46-s + (1.73 − i)53-s + (−0.866 − 0.5i)58-s − 64-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + i·8-s + (0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)16-s + (−0.866 + 0.499i)18-s − 0.999i·22-s + (0.866 + 0.5i)23-s + 29-s + (−0.866 − 0.5i)37-s − i·43-s + (−0.499 − 0.866i)46-s + (1.73 − i)53-s + (−0.866 − 0.5i)58-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.602 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.602 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6904748202\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6904748202\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 3 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + iT - T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.858218231977292076471550120816, −9.074730006505273925106632902293, −8.551994070807265698245022453178, −7.35417132820602436166106735253, −6.72996770311290153799569698002, −5.56296872112346410891689121620, −4.61555643021546928265907994248, −3.53465378423429131960271617445, −2.18037727613486672710851906101, −1.07470219746686778742392255655,
1.19080637093051140081446699663, 2.85617895297746116276147883246, 4.00195891924855911588036356450, 4.95337304366455609734181093077, 6.15832901537976846558656363582, 6.94733325653433445564612249417, 7.66611400817671502287097488321, 8.536869027870009288013885504184, 8.925033098753727908515378092148, 9.997958086047954778299508009591