L(s) = 1 | − i·7-s + 3·9-s − 11-s + 2i·13-s − 4i·17-s − 2·19-s − 5i·23-s − 29-s + 2·31-s − 3i·37-s + 12·41-s − 11i·43-s + 2i·47-s − 49-s + 6i·53-s + ⋯ |
L(s) = 1 | − 0.377i·7-s + 9-s − 0.301·11-s + 0.554i·13-s − 0.970i·17-s − 0.458·19-s − 1.04i·23-s − 0.185·29-s + 0.359·31-s − 0.493i·37-s + 1.87·41-s − 1.67i·43-s + 0.291i·47-s − 0.142·49-s + 0.824i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.730338566\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.730338566\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 - 3T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 5iT - 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 3iT - 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 + 11iT - 43T^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - iT - 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 9T + 79T^{2} \) |
| 83 | \( 1 - 2iT - 83T^{2} \) |
| 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
−8.767676850893497549172386042600, −7.74662939867041886646242846993, −7.19080763430036140943586240738, −6.53115139993931420625363737273, −5.57794553737165345132575280987, −4.51084993773464118217159838039, −4.15525470121499606203466314814, −2.88356868792995965327967952381, −1.90652841912408102382557785599, −0.61307819382894208145769915765,
1.18700357345533808297043422356, 2.22956998807617659933941189387, 3.32876956629738001565344348671, 4.20888597714119995921158889225, 5.02738150763096456837183848231, 5.94224177464664418584149948307, 6.56093804928038261803541032505, 7.64626743882928180275023609940, 7.983756260835750712005208834580, 8.987580220894569765943456026911