L(s) = 1 | + (1.73 + 1.73i)2-s + 3.99i·4-s + (−1.22 + 1.22i)7-s + (−3.46 + 3.46i)8-s − 4.24i·11-s + (3.67 + 3.67i)13-s − 4.24·14-s − 3.99·16-s + (−1.73 − 1.73i)17-s − 5i·19-s + (7.34 − 7.34i)22-s + (1.73 − 1.73i)23-s + 12.7i·26-s + (−4.89 − 4.89i)28-s − 4.24·29-s + ⋯ |
L(s) = 1 | + (1.22 + 1.22i)2-s + 1.99i·4-s + (−0.462 + 0.462i)7-s + (−1.22 + 1.22i)8-s − 1.27i·11-s + (1.01 + 1.01i)13-s − 1.13·14-s − 0.999·16-s + (−0.420 − 0.420i)17-s − 1.14i·19-s + (1.56 − 1.56i)22-s + (0.361 − 0.361i)23-s + 2.49i·26-s + (−0.925 − 0.925i)28-s − 0.787·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.296 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.26643 + 1.71930i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26643 + 1.71930i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-1.73 - 1.73i)T + 2iT^{2} \) |
| 7 | \( 1 + (1.22 - 1.22i)T - 7iT^{2} \) |
| 11 | \( 1 + 4.24iT - 11T^{2} \) |
| 13 | \( 1 + (-3.67 - 3.67i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.73 + 1.73i)T + 17iT^{2} \) |
| 19 | \( 1 + 5iT - 19T^{2} \) |
| 23 | \( 1 + (-1.73 + 1.73i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + (2.44 - 2.44i)T - 37iT^{2} \) |
| 41 | \( 1 + 8.48iT - 41T^{2} \) |
| 43 | \( 1 + (-1.22 - 1.22i)T + 43iT^{2} \) |
| 47 | \( 1 + (5.19 + 5.19i)T + 47iT^{2} \) |
| 53 | \( 1 + (6.92 - 6.92i)T - 53iT^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 + (3.67 - 3.67i)T - 67iT^{2} \) |
| 71 | \( 1 - 8.48iT - 71T^{2} \) |
| 73 | \( 1 + (2.44 + 2.44i)T + 73iT^{2} \) |
| 79 | \( 1 - 2iT - 79T^{2} \) |
| 83 | \( 1 + (-1.73 + 1.73i)T - 83iT^{2} \) |
| 89 | \( 1 + 8.48T + 89T^{2} \) |
| 97 | \( 1 + (-8.57 + 8.57i)T - 97iT^{2} \) |
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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
−12.98537422303710816203796328743, −11.80478786594968667772015138151, −10.98941226033724338146443290861, −9.126412882213425450545910360588, −8.470492976566357510671788378738, −7.05739459680011251421324783625, −6.32472411353171255543633717993, −5.43974171425160377335532104942, −4.19168673977928128978926271851, −3.03483342554141363842389643997,
1.66107925071076792843919169487, 3.26372502666141841769547144252, 4.15381994466599468141242269736, 5.37386830344088950947249805617, 6.46803442607228886238515453646, 7.937585008232601384281466879910, 9.587809450571857974593435357359, 10.34555405391366795293905913731, 11.08795902572862517851959309366, 12.15666584985147851265214895466