L(s) = 1 | + (2.32 + 4.01i)2-s + (−46.0 + 79.8i)3-s + (53.2 − 92.1i)4-s + (217. − 376. i)5-s − 427.·6-s + (19.1 − 33.0i)7-s + 1.08e3·8-s + (−3.15e3 − 5.46e3i)9-s + 2.01e3·10-s + 3.75e3·11-s + (4.90e3 + 8.50e3i)12-s + (−378. + 655. i)13-s + 177.·14-s + (2.00e4 + 3.47e4i)15-s + (−4.28e3 − 7.42e3i)16-s + (1.29e4 + 2.24e4i)17-s + ⋯ |
L(s) = 1 | + (0.205 + 0.355i)2-s + (−0.985 + 1.70i)3-s + (0.415 − 0.720i)4-s + (0.777 − 1.34i)5-s − 0.808·6-s + (0.0210 − 0.0364i)7-s + 0.751·8-s + (−1.44 − 2.49i)9-s + 0.637·10-s + 0.850·11-s + (0.819 + 1.41i)12-s + (−0.0477 + 0.0827i)13-s + 0.0172·14-s + (1.53 + 2.65i)15-s + (−0.261 − 0.453i)16-s + (0.640 + 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.118i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.992 - 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(1.80618 + 0.107295i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.80618 + 0.107295i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 + (-7.85e4 + 2.97e5i)T \) |
good | 2 | \( 1 + (-2.32 - 4.01i)T + (-64 + 110. i)T^{2} \) |
| 3 | \( 1 + (46.0 - 79.8i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-217. + 376. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-19.1 + 33.0i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 - 3.75e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (378. - 655. i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-1.29e4 - 2.24e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-2.28e4 + 3.95e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + 3.28e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 3.60e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.05e5T + 2.75e10T^{2} \) |
| 41 | \( 1 + (4.03e4 - 6.98e4i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 - 7.31e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.97e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (4.54e4 + 7.87e4i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (1.34e6 + 2.33e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (6.88e5 - 1.19e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (7.53e5 - 1.30e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-2.30e6 + 3.99e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + 3.68e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (2.38e6 - 4.12e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-3.58e6 - 6.20e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (2.55e5 + 4.42e5i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + 8.13e6T + 8.07e13T^{2} \) |
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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
−15.20655438213525552366032602472, −14.04070641634979357457778263056, −12.20650567416436741339416287721, −10.99482393118530213803677156047, −9.837010836113021528703308402614, −9.104085351288348498040017235211, −6.16693918458789849039186212247, −5.34321283030670572949835277343, −4.31848086107822193408893073955, −0.954379756841419345975421206755,
1.54391137394389228192634728050, 2.82676636745706362773448561416, 5.87387918529798938186677341235, 6.88484992769217331718634073250, 7.72927940052676681792878603982, 10.37729222932503743773300590646, 11.63373693463521635562827594927, 12.11360234846754738674423197193, 13.54994122620247694662870520568, 14.19088321169256072332068334356