L(s) = 1 | + (−0.772 − 1.55i)3-s + (2.60 + 0.477i)7-s + (−1.80 + 2.39i)9-s + (−1.34 + 0.773i)11-s + 4.18i·13-s + (−2.55 − 4.42i)17-s + (−4.62 − 2.67i)19-s + (−1.26 − 4.40i)21-s + (3.15 + 1.82i)23-s + (5.10 + 0.954i)27-s − 9.79i·29-s + (6.79 − 3.92i)31-s + (2.23 + 1.48i)33-s + (1.71 − 2.96i)37-s + (6.48 − 3.23i)39-s + ⋯ |
L(s) = 1 | + (−0.445 − 0.895i)3-s + (0.983 + 0.180i)7-s + (−0.602 + 0.798i)9-s + (−0.404 + 0.233i)11-s + 1.16i·13-s + (−0.619 − 1.07i)17-s + (−1.06 − 0.613i)19-s + (−0.276 − 0.960i)21-s + (0.658 + 0.380i)23-s + (0.982 + 0.183i)27-s − 1.81i·29-s + (1.22 − 0.705i)31-s + (0.389 + 0.257i)33-s + (0.281 − 0.488i)37-s + (1.03 − 0.517i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.409121006\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.409121006\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.772 + 1.55i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.60 - 0.477i)T \) |
good | 11 | \( 1 + (1.34 - 0.773i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 4.18iT - 13T^{2} \) |
| 17 | \( 1 + (2.55 + 4.42i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.62 + 2.67i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.15 - 1.82i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 9.79iT - 29T^{2} \) |
| 31 | \( 1 + (-6.79 + 3.92i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.71 + 2.96i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 8.82T + 41T^{2} \) |
| 43 | \( 1 - 3.79T + 43T^{2} \) |
| 47 | \( 1 + (1.24 - 2.16i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.684 - 0.395i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.73 + 4.73i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.76 + 3.90i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.58 - 9.67i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 7.97iT - 71T^{2} \) |
| 73 | \( 1 + (-10.7 + 6.19i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.12 + 1.94i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 5.26T + 83T^{2} \) |
| 89 | \( 1 + (-7.65 + 13.2i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 15.9iT - 97T^{2} \) |
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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
−8.874568758355985521468663523772, −7.943456267866024194043459978303, −7.45288679480943062501026564621, −6.56663749816540670129097708353, −5.92381459039148545855800065728, −4.76911609179131231452347508290, −4.42646358629767657601404728555, −2.53615030108421207227930360393, −2.04852372977860598773730176554, −0.62284269515201853617382207807,
1.06985416208464031276886379526, 2.58896265673921202010236145739, 3.65209372903049708836873725762, 4.53348485881994010197119854190, 5.17600644750021757340245701645, 5.97022182070066146331107789465, 6.79466695537021723757459925849, 8.070067440273755811495077299465, 8.403297092128204918208725396904, 9.242759155586900206346562164291