L(s) = 1 | + (1 + i)2-s + (0.133 − 0.5i)3-s + 2i·4-s + (−1.86 + 1.23i)5-s + (0.633 − 0.366i)6-s + (−0.866 + 2.5i)7-s + (−2 + 2i)8-s + (2.36 + 1.36i)9-s + (−3.09 − 0.633i)10-s + (2.36 − 1.36i)11-s + (1 + 0.267i)12-s + (−3 − 3i)13-s + (−3.36 + 1.63i)14-s + (0.366 + 1.09i)15-s − 4·16-s + (−1.09 + 4.09i)17-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + (0.0773 − 0.288i)3-s + i·4-s + (−0.834 + 0.550i)5-s + (0.258 − 0.149i)6-s + (−0.327 + 0.944i)7-s + (−0.707 + 0.707i)8-s + (0.788 + 0.455i)9-s + (−0.979 − 0.200i)10-s + (0.713 − 0.411i)11-s + (0.288 + 0.0773i)12-s + (−0.832 − 0.832i)13-s + (−0.899 + 0.436i)14-s + (0.0945 + 0.283i)15-s − 16-s + (−0.266 + 0.993i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.347 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.910917 + 1.30938i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.910917 + 1.30938i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - i)T \) |
| 5 | \( 1 + (1.86 - 1.23i)T \) |
| 7 | \( 1 + (0.866 - 2.5i)T \) |
good | 3 | \( 1 + (-0.133 + 0.5i)T + (-2.59 - 1.5i)T^{2} \) |
| 11 | \( 1 + (-2.36 + 1.36i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3 + 3i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.09 - 4.09i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.59 + 1.5i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 2.46T + 29T^{2} \) |
| 31 | \( 1 + (0.169 - 0.0980i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.73 + 0.464i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 6.46iT - 41T^{2} \) |
| 43 | \( 1 + (-0.633 + 0.633i)T - 43iT^{2} \) |
| 47 | \( 1 + (-8.83 + 2.36i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (11.1 + 3i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-4.26 + 2.46i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.96 - 3.40i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.13 - 7.96i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 8.53T + 71T^{2} \) |
| 73 | \( 1 + (10.8 + 2.90i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-0.803 - 0.464i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.56 - 7.56i)T + 83iT^{2} \) |
| 89 | \( 1 + (-3.40 + 5.89i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (9.73 + 9.73i)T + 97iT^{2} \) |
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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
−12.41284149295987391650113029350, −11.54518669762168911701888426422, −10.39419216567966950959691109409, −8.953287301961994851027625537752, −8.013393780779249849632467863231, −7.18891416598227618562113383834, −6.29252902301538921845250620458, −5.10573423713970040928633239350, −3.82254502998517373614979056898, −2.68270785133712729551857699180,
1.07820900346388312666851896048, 3.21494422213599849990499540359, 4.35721761208681542601523186705, 4.78478021936815025531800260754, 6.74540192067330739695182476977, 7.34015116366196970148201708593, 9.376209798241779414258866270483, 9.487760751155772441481019782998, 10.85354553709507605621061668946, 11.71208700042631971819077592899