L(s) = 1 | + 2i·2-s − 3i·3-s − 4·4-s + 6·6-s − 7i·7-s − 8i·8-s − 9·9-s − 8·11-s + 12i·12-s + 42i·13-s + 14·14-s + 16·16-s − 2i·17-s − 18i·18-s + 124·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.377i·7-s − 0.353i·8-s − 0.333·9-s − 0.219·11-s + 0.288i·12-s + 0.896i·13-s + 0.267·14-s + 0.250·16-s − 0.0285i·17-s − 0.235i·18-s + 1.49·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.636424954\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.636424954\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2iT \) |
| 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7iT \) |
good | 11 | \( 1 + 8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 42iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 2iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 124T + 6.85e3T^{2} \) |
| 23 | \( 1 + 76iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 254T + 2.43e4T^{2} \) |
| 31 | \( 1 + 72T + 2.97e4T^{2} \) |
| 37 | \( 1 - 398iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 462T + 6.89e4T^{2} \) |
| 43 | \( 1 + 212iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 264iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 162iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 772T + 2.05e5T^{2} \) |
| 61 | \( 1 - 30T + 2.26e5T^{2} \) |
| 67 | \( 1 + 764iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 236T + 3.57e5T^{2} \) |
| 73 | \( 1 + 418iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 552T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.03e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 30T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.19e3iT - 9.12e5T^{2} \) |
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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
−9.314182690133807997897260427468, −8.536387679335881130250575376501, −7.52105473003457330512499337081, −7.14794747389244718038993698401, −6.17161640814350353712642873615, −5.33537011117430618729079686259, −4.32689556998571317369229280244, −3.22187746258666948164001469507, −1.80397706923029321032773576506, −0.52923455932147020531116961815,
0.882158602390212355356864804421, 2.32680222745114737467175836851, 3.28909379588149547673159246492, 4.09880892651488697467796886898, 5.45743722133118799723398317999, 5.61589843217792366350753426585, 7.33954126303768993507721286438, 8.007495270772684071051343206614, 9.192250818553065540461622817942, 9.515296915587920759293439130122