| L(s) = 1 | − i·2-s − i·3-s − 4-s + (−2.18 + 0.489i)5-s − 6-s + i·8-s − 9-s + (0.489 + 2.18i)10-s − 2.39·11-s + i·12-s − 3.80i·13-s + (0.489 + 2.18i)15-s + 16-s + 1.97i·17-s + i·18-s − 4.17·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.975 + 0.218i)5-s − 0.408·6-s + 0.353i·8-s − 0.333·9-s + (0.154 + 0.689i)10-s − 0.721·11-s + 0.288i·12-s − 1.05i·13-s + (0.126 + 0.563i)15-s + 0.250·16-s + 0.477i·17-s + 0.235i·18-s − 0.956·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7376374666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7376374666\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.18 - 0.489i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2.39T + 11T^{2} \) |
| 13 | \( 1 + 3.80iT - 13T^{2} \) |
| 17 | \( 1 - 1.97iT - 17T^{2} \) |
| 19 | \( 1 + 4.17T + 19T^{2} \) |
| 23 | \( 1 - 8.17iT - 23T^{2} \) |
| 29 | \( 1 - 9.16T + 29T^{2} \) |
| 31 | \( 1 + 1.60T + 31T^{2} \) |
| 37 | \( 1 + 1.26iT - 37T^{2} \) |
| 41 | \( 1 - 3.19T + 41T^{2} \) |
| 43 | \( 1 - 4.90iT - 43T^{2} \) |
| 47 | \( 1 - 3.00iT - 47T^{2} \) |
| 53 | \( 1 - 12.1iT - 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 5.77iT - 67T^{2} \) |
| 71 | \( 1 - 6.94T + 71T^{2} \) |
| 73 | \( 1 - 0.506iT - 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 - 1.89iT - 83T^{2} \) |
| 89 | \( 1 - 1.97T + 89T^{2} \) |
| 97 | \( 1 - 12.0iT - 97T^{2} \) |
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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
−9.634128713849071991208957574435, −8.540477799716428629631720685142, −7.989911576925744662288555204084, −7.38150617679985717931221388572, −6.25668163112904316439872264996, −5.30870437098138714588342289722, −4.28571047751965818174235430164, −3.29425188451918978878209470550, −2.51691878980430057567867368238, −1.03854473994823635023887944558,
0.35876351151089061895929083400, 2.50026149690007607095461735652, 3.76733720670467530800166313206, 4.57079391750913747820594144342, 5.06144004602754803606359031973, 6.40455354779130944324265664732, 6.95584994088687399451685638593, 8.069992915472407222598592961856, 8.520046746628124322048754470351, 9.239006454361928970322018794164
