L(s) = 1 | + (−0.809 + 0.587i)2-s + (0.402 − 1.24i)3-s + (0.309 − 0.951i)4-s + (−0.529 − 2.17i)5-s + (0.402 + 1.24i)6-s + 1.35·7-s + (0.309 + 0.951i)8-s + (1.05 + 0.763i)9-s + (1.70 + 1.44i)10-s + (3.62 − 2.63i)11-s + (−1.05 − 0.766i)12-s + (−0.809 − 0.587i)13-s + (−1.09 + 0.797i)14-s + (−2.90 − 0.218i)15-s + (−0.809 − 0.587i)16-s + (−0.240 − 0.741i)17-s + ⋯ |
L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.232 − 0.716i)3-s + (0.154 − 0.475i)4-s + (−0.236 − 0.971i)5-s + (0.164 + 0.506i)6-s + 0.512·7-s + (0.109 + 0.336i)8-s + (0.350 + 0.254i)9-s + (0.539 + 0.457i)10-s + (1.09 − 0.793i)11-s + (−0.304 − 0.221i)12-s + (−0.224 − 0.163i)13-s + (−0.293 + 0.213i)14-s + (−0.750 − 0.0563i)15-s + (−0.202 − 0.146i)16-s + (−0.0584 − 0.179i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.285 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.04008 - 0.775029i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04008 - 0.775029i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + (0.529 + 2.17i)T \) |
| 13 | \( 1 + (0.809 + 0.587i)T \) |
good | 3 | \( 1 + (-0.402 + 1.24i)T + (-2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 - 1.35T + 7T^{2} \) |
| 11 | \( 1 + (-3.62 + 2.63i)T + (3.39 - 10.4i)T^{2} \) |
| 17 | \( 1 + (0.240 + 0.741i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.27 - 3.93i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.237 + 0.172i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.37 + 4.23i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.97 + 6.09i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (1.36 + 0.994i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.23 - 0.899i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + (-0.940 + 2.89i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (0.257 - 0.792i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-8.79 - 6.39i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-8.57 + 6.22i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (0.453 + 1.39i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (0.0974 - 0.299i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-3.32 + 2.41i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (1.52 - 4.69i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (0.124 + 0.382i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (1.61 - 1.17i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (3.59 - 11.0i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.13221310112092376436271237055, −9.327360414547896832387589000913, −8.331577182470067765428481209170, −8.011291049296932953212940013096, −7.01877375997737878392524859383, −6.01986996700109628583826365175, −4.99582476538459418546829792206, −3.85173633094992707023824868977, −1.94235512731990928251427232941, −0.922842975604042607017948965169,
1.65255369882136851953450475392, 3.08522632947596676965219227360, 3.97697569793222322221464496360, 4.93097988068201029420434752343, 6.78417042658217110104790821109, 7.03288464619898129622429360686, 8.337736151683470135901953217201, 9.208587027659313941559561900502, 9.900921657514684518694773531187, 10.57642103120635733816169729897