L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.499 + 0.866i)4-s + (−0.866 − 0.499i)6-s + (−1.5 + 2.59i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (−3.23 + 1.86i)11-s − 0.999i·12-s + (−3.5 − 0.866i)13-s − 3·14-s + (−0.5 − 0.866i)16-s + (3.46 + 2i)17-s + 0.999·18-s + (1.96 + 1.13i)19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.353 − 0.204i)6-s + (−0.566 + 0.981i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−0.974 + 0.562i)11-s − 0.288i·12-s + (−0.970 − 0.240i)13-s − 0.801·14-s + (−0.125 − 0.216i)16-s + (0.840 + 0.485i)17-s + 0.235·18-s + (0.450 + 0.260i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (3.5 + 0.866i)T \) |
good | 7 | \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.23 - 1.86i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.46 - 2i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.96 - 1.13i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 - 1.73i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.73 + 4.73i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 8.92iT - 31T^{2} \) |
| 37 | \( 1 + (-3.96 - 6.86i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.46 + 2i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.19 + 3i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 0.464T + 47T^{2} \) |
| 53 | \( 1 + 3.73iT - 53T^{2} \) |
| 59 | \( 1 + (3.92 + 2.26i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.73 + 6.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.73 + 4.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.803 + 0.464i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 + 2.53T + 83T^{2} \) |
| 89 | \( 1 + (-8.76 + 5.06i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.19 + 10.7i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.088404566221760335218294085596, −7.86615899918603924514607246153, −7.62941679773275284368331968723, −6.38811075358275037106605262539, −5.73734517382288182070797573711, −5.21583453749440559231278847871, −4.27835681103880114877619099807, −3.18043529260470390889008086896, −2.21132718150798341914396701885, 0,
1.16153817991010814552995330480, 2.61626270116687357620757998983, 3.39792750416127745301261692097, 4.49967556618138876296694104667, 5.26514233848340792891083683181, 6.01038533008779990631292437217, 7.15347445078652755021526338141, 7.47718836903496442007915264652, 8.652009813516757835381676438651