| L(s) = 1 | + (−18.3 + 31.7i)2-s + (82.9 + 143. i)3-s + (−416. − 720. i)4-s + (−705. − 1.22e3i)5-s − 6.08e3·6-s + (−777. − 1.34e3i)7-s + 1.17e4·8-s + (−3.91e3 + 6.77e3i)9-s + 5.17e4·10-s + 4.18e4·11-s + (6.90e4 − 1.19e5i)12-s + (−1.28e4 − 2.22e4i)13-s + 5.70e4·14-s + (1.17e5 − 2.02e5i)15-s + (−2.32e3 + 4.02e3i)16-s + (9.66e4 − 1.67e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.810 + 1.40i)2-s + (0.591 + 1.02i)3-s + (−0.812 − 1.40i)4-s + (−0.504 − 0.874i)5-s − 1.91·6-s + (−0.122 − 0.211i)7-s + 1.01·8-s + (−0.198 + 0.344i)9-s + 1.63·10-s + 0.861·11-s + (0.961 − 1.66i)12-s + (−0.124 − 0.216i)13-s + 0.396·14-s + (0.596 − 1.03i)15-s + (−0.00886 + 0.0153i)16-s + (0.280 − 0.486i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.551 - 0.833i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.551 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.02543 + 0.550956i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02543 + 0.550956i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (1.11e7 + 2.35e6i)T \) |
| good | 2 | \( 1 + (18.3 - 31.7i)T + (-256 - 443. i)T^{2} \) |
| 3 | \( 1 + (-82.9 - 143. i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (705. + 1.22e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 + (777. + 1.34e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 - 4.18e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (1.28e4 + 2.22e4i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 + (-9.66e4 + 1.67e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-8.10e4 - 1.40e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 - 5.22e4T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.33e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 7.89e5T + 2.64e13T^{2} \) |
| 41 | \( 1 + (1.22e7 + 2.11e7i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 + 3.44e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.48e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + (-1.20e7 + 2.08e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-6.45e7 + 1.11e8i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-5.44e7 - 9.43e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.34e8 - 2.33e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + (7.24e7 + 1.25e8i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 - 3.59e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + (-5.16e7 - 8.94e7i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-3.39e8 + 5.87e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (-1.07e8 + 1.85e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + 6.24e8T + 7.60e17T^{2} \) |
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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
−15.00679642949238276619459041357, −14.03197570292982389695268367299, −12.13592711985690993659350272472, −10.12734378521098416012117740891, −9.127358700838153193427626536509, −8.404430779255142504291213800021, −7.00729722564950192695281817290, −5.23624723503538305502200627939, −3.83204386733247676425832607835, −0.63346769083489453819346084561,
1.17675078966816221753981228363, 2.41174441546399436995589932237, 3.58513571909232992848058107743, 6.77748240903123430277920950286, 8.024318261581267863500914113298, 9.207196636591519462723851200125, 10.60141350525364867181816694156, 11.70744316669633317942526101032, 12.57849407341925813107267280410, 13.85885721228053480464193042515
