| L(s) = 1 | + (1.91 − 0.0910i)2-s + (1.65 − 0.509i)3-s + (1.65 − 0.158i)4-s + (−1.87 − 1.78i)5-s + (3.11 − 1.12i)6-s + (0.424 + 2.20i)7-s + (−0.637 + 0.0916i)8-s + (2.48 − 1.68i)9-s + (−3.74 − 3.24i)10-s + (−4.26 + 2.19i)11-s + (2.66 − 1.10i)12-s + (3.61 + 0.696i)13-s + (1.01 + 4.17i)14-s + (−4.01 − 2.00i)15-s + (−4.47 + 0.862i)16-s + (0.858 + 1.87i)17-s + ⋯ |
| L(s) = 1 | + (1.35 − 0.0643i)2-s + (0.955 − 0.294i)3-s + (0.827 − 0.0790i)4-s + (−0.838 − 0.799i)5-s + (1.27 − 0.459i)6-s + (0.160 + 0.832i)7-s + (−0.225 + 0.0324i)8-s + (0.826 − 0.562i)9-s + (−1.18 − 1.02i)10-s + (−1.28 + 0.662i)11-s + (0.768 − 0.319i)12-s + (1.00 + 0.193i)13-s + (0.270 + 1.11i)14-s + (−1.03 − 0.517i)15-s + (−1.11 + 0.215i)16-s + (0.208 + 0.455i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.47201 - 0.490237i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.47201 - 0.490237i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.65 + 0.509i)T \) |
| 23 | \( 1 + (0.573 + 4.76i)T \) |
| good | 2 | \( 1 + (-1.91 + 0.0910i)T + (1.99 - 0.190i)T^{2} \) |
| 5 | \( 1 + (1.87 + 1.78i)T + (0.237 + 4.99i)T^{2} \) |
| 7 | \( 1 + (-0.424 - 2.20i)T + (-6.49 + 2.60i)T^{2} \) |
| 11 | \( 1 + (4.26 - 2.19i)T + (6.38 - 8.96i)T^{2} \) |
| 13 | \( 1 + (-3.61 - 0.696i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (-0.858 - 1.87i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (0.409 + 0.187i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (0.657 - 6.88i)T + (-28.4 - 5.48i)T^{2} \) |
| 31 | \( 1 + (2.87 - 2.25i)T + (7.30 - 30.1i)T^{2} \) |
| 37 | \( 1 + (-1.58 + 5.39i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-6.57 + 6.89i)T + (-1.95 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-3.91 + 4.98i)T + (-10.1 - 41.7i)T^{2} \) |
| 47 | \( 1 + (8.90 + 5.13i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.890 - 1.02i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-0.375 + 1.94i)T + (-54.7 - 21.9i)T^{2} \) |
| 61 | \( 1 + (5.07 - 12.6i)T + (-44.1 - 42.0i)T^{2} \) |
| 67 | \( 1 + (-0.445 + 0.864i)T + (-38.8 - 54.5i)T^{2} \) |
| 71 | \( 1 + (1.01 + 1.57i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (5.00 - 10.9i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-6.31 + 2.18i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (7.37 - 7.03i)T + (3.94 - 82.9i)T^{2} \) |
| 89 | \( 1 + (-2.09 + 14.5i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-3.16 - 0.768i)T + (86.2 + 44.4i)T^{2} \) |
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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
−12.68075016279508411089530745401, −11.98463346042085617267739947882, −10.63146600654926412456720698967, −8.929290858322258575439899443543, −8.462149274804545598539851125960, −7.24628901772808264191560927397, −5.74419366084488535385577492090, −4.63947972030040267410664476081, −3.65976752464577311816621559384, −2.31912401035301126267722924100,
2.93725705810328785284474942726, 3.62054303068549525907401127118, 4.62874605036612706994850037855, 6.06376928490764533366149660619, 7.49761830916720679324481951334, 8.060299666138439331995635162097, 9.612149755564258625341386651914, 10.87321622830654033362068759463, 11.41284242041275420891161619438, 12.99373059468842633690168612353
