| L(s) = 1 | + (−22.6 + 39.1i)2-s + (745. − 430. i)3-s + (−1.02e3 − 1.77e3i)4-s + (7.53e3 + 4.34e3i)5-s + 3.89e4i·6-s + 9.26e4·8-s + (1.05e5 − 1.82e5i)9-s + (−3.40e5 + 1.96e5i)10-s + (2.57e5 + 4.46e5i)11-s + (−1.52e6 − 8.82e5i)12-s − 2.74e6i·13-s + 7.49e6·15-s + (−2.09e6 + 3.63e6i)16-s + (−5.50e6 + 3.17e6i)17-s + (4.76e6 + 8.25e6i)18-s + (−6.71e7 − 3.87e7i)19-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (1.02 − 0.590i)3-s + (−0.249 − 0.433i)4-s + (0.482 + 0.278i)5-s + 0.835i·6-s + 0.353·8-s + (0.198 − 0.343i)9-s + (−0.340 + 0.196i)10-s + (0.145 + 0.252i)11-s + (−0.511 − 0.295i)12-s − 0.569i·13-s + 0.657·15-s + (−0.125 + 0.216i)16-s + (−0.228 + 0.131i)17-s + (0.140 + 0.242i)18-s + (−1.42 − 0.824i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(1.828458161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.828458161\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (22.6 - 39.1i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (-745. + 430. i)T + (2.65e5 - 4.60e5i)T^{2} \) |
| 5 | \( 1 + (-7.53e3 - 4.34e3i)T + (1.22e8 + 2.11e8i)T^{2} \) |
| 11 | \( 1 + (-2.57e5 - 4.46e5i)T + (-1.56e12 + 2.71e12i)T^{2} \) |
| 13 | \( 1 + 2.74e6iT - 2.32e13T^{2} \) |
| 17 | \( 1 + (5.50e6 - 3.17e6i)T + (2.91e14 - 5.04e14i)T^{2} \) |
| 19 | \( 1 + (6.71e7 + 3.87e7i)T + (1.10e15 + 1.91e15i)T^{2} \) |
| 23 | \( 1 + (-5.20e7 + 9.01e7i)T + (-1.09e16 - 1.89e16i)T^{2} \) |
| 29 | \( 1 + 1.41e8T + 3.53e17T^{2} \) |
| 31 | \( 1 + (-6.99e8 + 4.03e8i)T + (3.93e17 - 6.82e17i)T^{2} \) |
| 37 | \( 1 + (-1.86e9 + 3.23e9i)T + (-3.29e18 - 5.70e18i)T^{2} \) |
| 41 | \( 1 - 4.75e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 - 8.94e9T + 3.99e19T^{2} \) |
| 47 | \( 1 + (1.79e10 + 1.03e10i)T + (5.80e19 + 1.00e20i)T^{2} \) |
| 53 | \( 1 + (1.17e10 + 2.03e10i)T + (-2.45e20 + 4.25e20i)T^{2} \) |
| 59 | \( 1 + (-1.83e9 + 1.05e9i)T + (8.89e20 - 1.54e21i)T^{2} \) |
| 61 | \( 1 + (-7.28e9 - 4.20e9i)T + (1.32e21 + 2.29e21i)T^{2} \) |
| 67 | \( 1 + (-3.27e10 - 5.66e10i)T + (-4.09e21 + 7.08e21i)T^{2} \) |
| 71 | \( 1 + 2.46e11T + 1.64e22T^{2} \) |
| 73 | \( 1 + (8.87e10 - 5.12e10i)T + (1.14e22 - 1.98e22i)T^{2} \) |
| 79 | \( 1 + (-1.38e11 + 2.39e11i)T + (-2.95e22 - 5.11e22i)T^{2} \) |
| 83 | \( 1 + 8.18e10iT - 1.06e23T^{2} \) |
| 89 | \( 1 + (-4.69e10 - 2.71e10i)T + (1.23e23 + 2.13e23i)T^{2} \) |
| 97 | \( 1 + 3.42e11iT - 6.93e23T^{2} \) |
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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
−11.05264222738873635246344416794, −9.935302286849829347803449062750, −8.827240546971638448707909085858, −8.073912348151967725435586459157, −6.99658802530835223837204537509, −6.03173151057027338441717000440, −4.45769558173488548252293386715, −2.75062875352172611373039577222, −1.88697294975911433770164479620, −0.37629454217047008012829895771,
1.33739291272494323533001495335, 2.42467500252553449920091676656, 3.55244999042846088182627086405, 4.55940120661957179884605630513, 6.23090086308045222773492855868, 7.899927556693571055260493651479, 8.891136456462613573713077928057, 9.488022578193756677040308151742, 10.48686314556817454183321893605, 11.66413018411508628690925995876
