| L(s) = 1 | + 2i·2-s − 3i·3-s − 4·4-s + 6·6-s − 7i·7-s − 8i·8-s − 9·9-s + 12·11-s + 12i·12-s + 2i·13-s + 14·14-s + 16·16-s + 18i·17-s − 18i·18-s − 56·19-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 + 0.328·11-s + 0.288i·12-s + 0.0426i·13-s + 0.267·14-s + 0.250·16-s + 0.256i·17-s − 0.235i·18-s − 0.676·19-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
| \(L(2)\) |
\(\approx\) |
\(0.3243440001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3243440001\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 12T + 1.33e3T^{2} \) |
| 13 | \( 1 - 2iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 18iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 56T + 6.85e3T^{2} \) |
| 23 | \( 1 + 156iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 186T + 2.43e4T^{2} \) |
| 31 | \( 1 + 52T + 2.97e4T^{2} \) |
| 37 | \( 1 - 178iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 138T + 6.89e4T^{2} \) |
| 43 | \( 1 + 412iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 456iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 198iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 348T + 2.05e5T^{2} \) |
| 61 | \( 1 - 110T + 2.26e5T^{2} \) |
| 67 | \( 1 - 196iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 936T + 3.57e5T^{2} \) |
| 73 | \( 1 - 542iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 992T + 4.93e5T^{2} \) |
| 83 | \( 1 + 276iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 630T + 7.04e5T^{2} \) |
| 97 | \( 1 + 110iT - 9.12e5T^{2} \) |
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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
−8.795000050024085728937762306747, −8.380054645196793331059942843571, −7.38858626036504869416919771435, −6.64366653768123116711881353579, −6.06412766728581818853408996045, −4.86985616512865320848900805764, −4.03250376015538611714192650433, −2.72121574952047424550454837171, −1.32887792751900425458677793114, −0.082601983793542846395749725795,
1.45027622095930457588113336544, 2.66795041368607770297432933058, 3.60379600779378439016063006612, 4.53004213511625272898821741486, 5.40809973052758137815999405038, 6.34242491535372902634093066242, 7.54331317735937527499687633393, 8.550447342069368825522227885318, 9.187410194715015616685125333440, 9.926459559057377343209621466910
