L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s − 3i·7-s + i·8-s − 9-s − 5·11-s + i·12-s − 2i·13-s − 3·14-s + 16-s − i·17-s + i·18-s − 19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s − 1.13i·7-s + 0.353i·8-s − 0.333·9-s − 1.50·11-s + 0.288i·12-s − 0.554i·13-s − 0.801·14-s + 0.250·16-s − 0.242i·17-s + 0.235i·18-s − 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 + iT \) |
good | 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 - 3iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + iT - 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 + iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 11iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 - 18iT - 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 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
−7.977376926333309118175734280291, −7.74454738505994152014273527758, −6.94274765063363805424984972340, −5.75379687540167972938753962772, −5.13243726669882680950299969767, −4.07853809484330157121327256064, −3.20650849535453936657773173674, −2.31404446576434806501420573190, −1.14469489677111027080542182623, 0,
2.15227897478228432824090523678, 2.98209019074261041681377611880, 4.21844450252741656683311538202, 4.92920505331257891477187201512, 5.71549906655029950639976440705, 6.20519285789021037069340056440, 7.34613799676728884530216814158, 8.062297722908699188937248350481, 8.753085849492145649410358642210