| L(s) = 1 | + (1.36 + 0.379i)2-s + (−0.453 − 0.891i)3-s + (1.71 + 1.03i)4-s + (−1.79 + 1.33i)5-s + (−0.280 − 1.38i)6-s + (2.79 + 2.79i)7-s + (1.94 + 2.05i)8-s + (−0.587 + 0.809i)9-s + (−2.94 + 1.14i)10-s + (1.49 + 2.06i)11-s + (0.143 − 1.99i)12-s + (−1.10 − 6.98i)13-s + (2.74 + 4.86i)14-s + (2.00 + 0.989i)15-s + (1.86 + 3.53i)16-s + (2.15 + 1.09i)17-s + ⋯ |
| L(s) = 1 | + (0.963 + 0.268i)2-s + (−0.262 − 0.514i)3-s + (0.856 + 0.516i)4-s + (−0.801 + 0.598i)5-s + (−0.114 − 0.565i)6-s + (1.05 + 1.05i)7-s + (0.686 + 0.727i)8-s + (−0.195 + 0.269i)9-s + (−0.932 + 0.361i)10-s + (0.451 + 0.621i)11-s + (0.0414 − 0.575i)12-s + (−0.306 − 1.93i)13-s + (0.734 + 1.30i)14-s + (0.517 + 0.255i)15-s + (0.465 + 0.884i)16-s + (0.522 + 0.266i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.92616 + 0.711727i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.92616 + 0.711727i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.36 - 0.379i)T \) |
| 3 | \( 1 + (0.453 + 0.891i)T \) |
| 5 | \( 1 + (1.79 - 1.33i)T \) |
| good | 7 | \( 1 + (-2.79 - 2.79i)T + 7iT^{2} \) |
| 11 | \( 1 + (-1.49 - 2.06i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (1.10 + 6.98i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-2.15 - 1.09i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (1.18 + 3.63i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.422 - 2.66i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (3.49 + 1.13i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.552 - 0.179i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (6.38 - 1.01i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (9.43 + 6.85i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-2.64 + 2.64i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.64 + 1.85i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-1.48 + 0.757i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-3.32 - 2.41i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.266 + 0.193i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-6.22 + 12.2i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (13.4 + 4.38i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-10.3 - 1.63i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (1.99 - 6.13i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-11.3 - 5.79i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (5.98 + 8.24i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.0710 - 0.139i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−12.11741309692591652885436299427, −11.25253422973615723014286858710, −10.43908758066942609349171377953, −8.518515455332097872899423916849, −7.75804955382946732625775028902, −7.00816223430160184639558388132, −5.68910600536093597138362347399, −4.98834388828857764097275205043, −3.46733466359960598520427721471, −2.19704425122081502258035036688,
1.46474808310859436528076646611, 3.75512814772753236340775367705, 4.31875363883147210716443722922, 5.18410182240242194892527575023, 6.63179535035667376009989487145, 7.61157237460806760955584659464, 8.809979453796751635561283682716, 10.07265714049001650986272262073, 11.11965373596562735449927719497, 11.60224194397380073941382858325
