L(s) = 1 | − i·2-s − 3-s − 4-s + 3i·5-s + i·6-s − 3i·7-s + i·8-s − 2·9-s + 3·10-s + 12-s + (2 − 3i)13-s − 3·14-s − 3i·15-s + 16-s + 3·17-s + 2i·18-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577·3-s − 0.5·4-s + 1.34i·5-s + 0.408i·6-s − 1.13i·7-s + 0.353i·8-s − 0.666·9-s + 0.948·10-s + 0.288·12-s + (0.554 − 0.832i)13-s − 0.801·14-s − 0.774i·15-s + 0.250·16-s + 0.727·17-s + 0.471i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.538704 - 0.163106i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.538704 - 0.163106i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 13 | \( 1 + (-2 + 3i)T \) |
good | 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 - 3iT - 5T^{2} \) |
| 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 3iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 15iT - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−17.69518615041136556702863141543, −16.49082166238566739681079280254, −14.62874621167701846399328933010, −13.77322524224678752979164692856, −12.06840841229674551327782820165, −10.78604730673694910604982598109, −10.24097835334179608955390309155, −7.81580411016520448253038462683, −6.01632195233972618102175746207, −3.49748557902694879901443144890,
4.94931671065091800429494720066, 6.10272337101188958807676622402, 8.380034989348270758343219773095, 9.273169857449595050678731052098, 11.58320108030543861280167783156, 12.56955955660686972306929715682, 14.00419847866366404346413565835, 15.59041563034695015036545603647, 16.44997075291564349503830075472, 17.36882624358461094367934412757