| L(s) = 1 | + (11.7 + 11.7i)2-s + 78.9i·3-s + 21.8i·4-s + (740. − 740. i)5-s + (−930. + 930. i)6-s + 1.53e3·7-s + (2.75e3 − 2.75e3i)8-s + 334.·9-s + 1.74e4·10-s − 1.88e4i·11-s − 1.72e3·12-s + (−8.07e3 + 8.07e3i)13-s + (1.81e4 + 1.81e4i)14-s + (5.83e4 + 5.83e4i)15-s + 7.06e4·16-s + (−7.72e4 + 7.72e4i)17-s + ⋯ |
| L(s) = 1 | + (0.736 + 0.736i)2-s + 0.974i·3-s + 0.0853i·4-s + (1.18 − 1.18i)5-s + (−0.717 + 0.717i)6-s + 0.639·7-s + (0.673 − 0.673i)8-s + 0.0509·9-s + 1.74·10-s − 1.28i·11-s − 0.0831·12-s + (−0.282 + 0.282i)13-s + (0.471 + 0.471i)14-s + (1.15 + 1.15i)15-s + 1.07·16-s + (−0.924 + 0.924i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.38534 + 1.11072i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.38534 + 1.11072i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-4.34e5 + 1.82e6i)T \) |
| good | 2 | \( 1 + (-11.7 - 11.7i)T + 256iT^{2} \) |
| 3 | \( 1 - 78.9iT - 6.56e3T^{2} \) |
| 5 | \( 1 + (-740. + 740. i)T - 3.90e5iT^{2} \) |
| 7 | \( 1 - 1.53e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 1.88e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + (8.07e3 - 8.07e3i)T - 8.15e8iT^{2} \) |
| 17 | \( 1 + (7.72e4 - 7.72e4i)T - 6.97e9iT^{2} \) |
| 19 | \( 1 + (3.41e4 - 3.41e4i)T - 1.69e10iT^{2} \) |
| 23 | \( 1 + (5.00e4 - 5.00e4i)T - 7.83e10iT^{2} \) |
| 29 | \( 1 + (1.09e4 + 1.09e4i)T + 5.00e11iT^{2} \) |
| 31 | \( 1 + (-7.88e5 - 7.88e5i)T + 8.52e11iT^{2} \) |
| 41 | \( 1 - 3.51e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + (-5.56e5 + 5.56e5i)T - 1.16e13iT^{2} \) |
| 47 | \( 1 - 4.96e5T + 2.38e13T^{2} \) |
| 53 | \( 1 + 1.01e7T + 6.22e13T^{2} \) |
| 59 | \( 1 + (7.20e5 - 7.20e5i)T - 1.46e14iT^{2} \) |
| 61 | \( 1 + (-6.21e6 - 6.21e6i)T + 1.91e14iT^{2} \) |
| 67 | \( 1 - 8.75e6iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 1.28e7T + 6.45e14T^{2} \) |
| 73 | \( 1 - 5.22e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + (-5.36e7 + 5.36e7i)T - 1.51e15iT^{2} \) |
| 83 | \( 1 + 2.23e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + (-4.99e7 - 4.99e7i)T + 3.93e15iT^{2} \) |
| 97 | \( 1 + (1.06e8 - 1.06e8i)T - 7.83e15iT^{2} \) |
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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
−14.69984309040611732317045792641, −13.74320811746004985975407290264, −12.83025716884654697041566944942, −10.82069511594136598234273815528, −9.673199770674442181808816249262, −8.481021475606265094264109973980, −6.21751884368773309801468695671, −5.17365648841145029203938859799, −4.25678383982598666207427442322, −1.40499211697608777248050101526,
1.85988078013882410719260312145, 2.54122729784599400971511997695, 4.73747312956001267974098155267, 6.59144743891027324931230708074, 7.63973697019140106773850367673, 9.866634835939984132117372475737, 11.06459628852339382263436866781, 12.25061824680809396061287311625, 13.31935949755045543189198316226, 13.99835589518615282376699152609
