L(s) = 1 | − i·2-s − 4-s + 4i·7-s + i·8-s − 3·11-s − i·13-s + 4·14-s + 16-s − 2·19-s + 3i·22-s − 9i·23-s − 26-s − 4i·28-s − 6·29-s − 10·31-s − i·32-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 1.51i·7-s + 0.353i·8-s − 0.904·11-s − 0.277i·13-s + 1.06·14-s + 0.250·16-s − 0.458·19-s + 0.639i·22-s − 1.87i·23-s − 0.196·26-s − 0.755i·28-s − 1.11·29-s − 1.79·31-s − 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 9iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 - 7iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 15T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 - 19iT - 97T^{2} \) |
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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
−8.969807269776292482679102854866, −8.678601480541626286136654443722, −7.75135484508714870051209006322, −6.55914519094423828045155233596, −5.52439635985594017990594264392, −5.04250681498774848020633839555, −3.74921696363360829586538440219, −2.64802911163887194713286263259, −2.00805247241882622671154375703, 0,
1.64119946306213127505685108561, 3.39469069070105949874617725048, 4.12813381427571540358976871030, 5.14095944070439049398360999604, 5.93172244312131361664149722274, 7.11732366715655303509778126209, 7.44610568031542127505266294045, 8.187238228461980480405173279197, 9.358767197798941468871957599282