| L(s) = 1 | + (−0.769 + 3.36i)2-s + (18.8 − 23.5i)3-s + (18.0 + 8.70i)4-s + (−1.58 + 1.98i)5-s + (64.9 + 81.4i)6-s + (112. + 63.8i)7-s + (−112. + 140. i)8-s + (−148. − 649. i)9-s + (−5.47 − 6.86i)10-s + (92.7 − 406. i)11-s + (544. − 262. i)12-s + (−97.0 + 425. i)13-s + (−302. + 330. i)14-s + (17.0 + 74.6i)15-s + (12.3 + 15.5i)16-s + (205. − 98.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.135 + 0.595i)2-s + (1.20 − 1.51i)3-s + (0.564 + 0.271i)4-s + (−0.0283 + 0.0355i)5-s + (0.736 + 0.924i)6-s + (0.870 + 0.492i)7-s + (−0.619 + 0.777i)8-s + (−0.610 − 2.67i)9-s + (−0.0173 − 0.0217i)10-s + (0.231 − 1.01i)11-s + (1.09 − 0.525i)12-s + (−0.159 + 0.697i)13-s + (−0.411 + 0.451i)14-s + (0.0195 + 0.0857i)15-s + (0.0120 + 0.0151i)16-s + (0.172 − 0.0830i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.60987 - 0.428400i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.60987 - 0.428400i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-112. - 63.8i)T \) |
| good | 2 | \( 1 + (0.769 - 3.36i)T + (-28.8 - 13.8i)T^{2} \) |
| 3 | \( 1 + (-18.8 + 23.5i)T + (-54.0 - 236. i)T^{2} \) |
| 5 | \( 1 + (1.58 - 1.98i)T + (-695. - 3.04e3i)T^{2} \) |
| 11 | \( 1 + (-92.7 + 406. i)T + (-1.45e5 - 6.98e4i)T^{2} \) |
| 13 | \( 1 + (97.0 - 425. i)T + (-3.34e5 - 1.61e5i)T^{2} \) |
| 17 | \( 1 + (-205. + 98.9i)T + (8.85e5 - 1.11e6i)T^{2} \) |
| 19 | \( 1 - 388.T + 2.47e6T^{2} \) |
| 23 | \( 1 + (1.55e3 + 747. i)T + (4.01e6 + 5.03e6i)T^{2} \) |
| 29 | \( 1 + (-5.79e3 + 2.79e3i)T + (1.27e7 - 1.60e7i)T^{2} \) |
| 31 | \( 1 + 4.57e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (8.79e3 - 4.23e3i)T + (4.32e7 - 5.42e7i)T^{2} \) |
| 41 | \( 1 + (8.82e3 - 1.10e4i)T + (-2.57e7 - 1.12e8i)T^{2} \) |
| 43 | \( 1 + (407. + 510. i)T + (-3.27e7 + 1.43e8i)T^{2} \) |
| 47 | \( 1 + (1.72e3 - 7.55e3i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (-2.75e4 - 1.32e4i)T + (2.60e8 + 3.26e8i)T^{2} \) |
| 59 | \( 1 + (1.67e4 + 2.09e4i)T + (-1.59e8 + 6.96e8i)T^{2} \) |
| 61 | \( 1 + (2.57e4 - 1.23e4i)T + (5.26e8 - 6.60e8i)T^{2} \) |
| 67 | \( 1 + 6.12e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-6.41e3 - 3.09e3i)T + (1.12e9 + 1.41e9i)T^{2} \) |
| 73 | \( 1 + (-1.44e4 - 6.32e4i)T + (-1.86e9 + 8.99e8i)T^{2} \) |
| 79 | \( 1 - 5.87e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (1.16e4 + 5.09e4i)T + (-3.54e9 + 1.70e9i)T^{2} \) |
| 89 | \( 1 + (2.65e3 + 1.16e4i)T + (-5.03e9 + 2.42e9i)T^{2} \) |
| 97 | \( 1 - 6.23e4T + 8.58e9T^{2} \) |
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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
−14.43995961931609564719580264556, −13.71185285927455396140069127260, −12.22062238607749508320478062455, −11.52134840320358341674547517893, −8.902670499332498599292336398498, −8.208512775677866691164420835800, −7.21686291687271365966708688637, −6.08762185968488153253183612443, −3.01984528309524794183117852345, −1.65344670648041913880489723301,
2.08017723582652349140328208601, 3.59950665588435714778700458200, 4.97721355057255096857575508827, 7.53684580392265260167706112712, 8.880927524214957920220038423278, 10.22292009626684936051673979394, 10.52755885553741487294992461896, 12.07037408956251667858939343480, 13.91365922247543091623247629797, 14.81573591571966596008223275279
