L(s) = 1 | + (1.90 − 1.90i)2-s + (−1.68 − 0.394i)3-s − 5.22i·4-s + (−3.95 + 2.45i)6-s + (−0.707 − 0.707i)7-s + (−6.13 − 6.13i)8-s + (2.68 + 1.33i)9-s − 3.76i·11-s + (−2.06 + 8.81i)12-s + (−3.48 + 3.48i)13-s − 2.68·14-s − 12.8·16-s + (−0.131 + 0.131i)17-s + (7.64 − 2.57i)18-s − 3.89i·19-s + ⋯ |
L(s) = 1 | + (1.34 − 1.34i)2-s + (−0.973 − 0.227i)3-s − 2.61i·4-s + (−1.61 + 1.00i)6-s + (−0.267 − 0.267i)7-s + (−2.16 − 2.16i)8-s + (0.896 + 0.443i)9-s − 1.13i·11-s + (−0.595 + 2.54i)12-s + (−0.965 + 0.965i)13-s − 0.718·14-s − 3.21·16-s + (−0.0319 + 0.0319i)17-s + (1.80 − 0.608i)18-s − 0.893i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.311877 + 1.70880i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.311877 + 1.70880i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.68 + 0.394i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 2 | \( 1 + (-1.90 + 1.90i)T - 2iT^{2} \) |
| 11 | \( 1 + 3.76iT - 11T^{2} \) |
| 13 | \( 1 + (3.48 - 3.48i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.131 - 0.131i)T - 17iT^{2} \) |
| 19 | \( 1 + 3.89iT - 19T^{2} \) |
| 23 | \( 1 + (-3.35 - 3.35i)T + 23iT^{2} \) |
| 29 | \( 1 - 4.27T + 29T^{2} \) |
| 31 | \( 1 - 3.35T + 31T^{2} \) |
| 37 | \( 1 + (4.98 + 4.98i)T + 37iT^{2} \) |
| 41 | \( 1 - 1.16iT - 41T^{2} \) |
| 43 | \( 1 + (-2.05 + 2.05i)T - 43iT^{2} \) |
| 47 | \( 1 + (-7.97 + 7.97i)T - 47iT^{2} \) |
| 53 | \( 1 + (3.80 + 3.80i)T + 53iT^{2} \) |
| 59 | \( 1 - 7.06T + 59T^{2} \) |
| 61 | \( 1 + 4.11T + 61T^{2} \) |
| 67 | \( 1 + (-0.153 - 0.153i)T + 67iT^{2} \) |
| 71 | \( 1 + 2.12iT - 71T^{2} \) |
| 73 | \( 1 + (-9.79 + 9.79i)T - 73iT^{2} \) |
| 79 | \( 1 + 0.147iT - 79T^{2} \) |
| 83 | \( 1 + (-2.58 - 2.58i)T + 83iT^{2} \) |
| 89 | \( 1 + 1.17T + 89T^{2} \) |
| 97 | \( 1 + (1.52 + 1.52i)T + 97iT^{2} \) |
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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
−10.79055326791941523138414427413, −10.00019664697042368572586707172, −9.055721137216577702300533080590, −7.12099719468273807249986200313, −6.32746436020034225312011175692, −5.33537053417681525346095771815, −4.63176260094985715752263050715, −3.54398158724780404821288537667, −2.25085064223077355487828633082, −0.74773542033508907979720793639,
2.86070156259727568788427048359, 4.25288710839373740573218894013, 4.94570450070001331536155347263, 5.70789218591626251898804731619, 6.62571300362827006906150103534, 7.29051792174118497568660283688, 8.222186907172267865729445508739, 9.602365853337292633807314501465, 10.54779855709852229176311080949, 11.89920397911549797526085535174