L(s) = 1 | + (−0.167 + 0.167i)2-s + (0.707 − 0.707i)3-s + 1.94i·4-s + (2.23 − 0.0836i)5-s + 0.236i·6-s + (−2.64 + 0.0627i)7-s + (−0.658 − 0.658i)8-s − 1.00i·9-s + (−0.359 + 0.387i)10-s + 3.98·11-s + (1.37 + 1.37i)12-s + (0.500 − 0.500i)13-s + (0.431 − 0.452i)14-s + (1.52 − 1.63i)15-s − 3.66·16-s + (−1.67 − 1.67i)17-s + ⋯ |
L(s) = 1 | + (−0.118 + 0.118i)2-s + (0.408 − 0.408i)3-s + 0.972i·4-s + (0.999 − 0.0373i)5-s + 0.0964i·6-s + (−0.999 + 0.0237i)7-s + (−0.232 − 0.232i)8-s − 0.333i·9-s + (−0.113 + 0.122i)10-s + 1.20·11-s + (0.396 + 0.396i)12-s + (0.138 − 0.138i)13-s + (0.115 − 0.120i)14-s + (0.392 − 0.423i)15-s − 0.917·16-s + (−0.407 − 0.407i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.12400 + 0.165813i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.12400 + 0.165813i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (-2.23 + 0.0836i)T \) |
| 7 | \( 1 + (2.64 - 0.0627i)T \) |
good | 2 | \( 1 + (0.167 - 0.167i)T - 2iT^{2} \) |
| 11 | \( 1 - 3.98T + 11T^{2} \) |
| 13 | \( 1 + (-0.500 + 0.500i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.67 + 1.67i)T + 17iT^{2} \) |
| 19 | \( 1 + 7.21T + 19T^{2} \) |
| 23 | \( 1 + (5.16 + 5.16i)T + 23iT^{2} \) |
| 29 | \( 1 - 3.65iT - 29T^{2} \) |
| 31 | \( 1 - 4.93iT - 31T^{2} \) |
| 37 | \( 1 + (-0.292 + 0.292i)T - 37iT^{2} \) |
| 41 | \( 1 + 7.63iT - 41T^{2} \) |
| 43 | \( 1 + (-3.65 - 3.65i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.305 + 0.305i)T + 47iT^{2} \) |
| 53 | \( 1 + (-5.39 - 5.39i)T + 53iT^{2} \) |
| 59 | \( 1 - 6.10T + 59T^{2} \) |
| 61 | \( 1 - 7.11iT - 61T^{2} \) |
| 67 | \( 1 + (-0.944 + 0.944i)T - 67iT^{2} \) |
| 71 | \( 1 - 1.19T + 71T^{2} \) |
| 73 | \( 1 + (-1.38 + 1.38i)T - 73iT^{2} \) |
| 79 | \( 1 + 8.64iT - 79T^{2} \) |
| 83 | \( 1 + (11.9 - 11.9i)T - 83iT^{2} \) |
| 89 | \( 1 - 7.82T + 89T^{2} \) |
| 97 | \( 1 + (-7.43 - 7.43i)T + 97iT^{2} \) |
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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
−13.67181133343416460927075777799, −12.78444056436345442803068951196, −12.16485668597509931507013604426, −10.49088985786540512219479063112, −9.175543851258481321410862755918, −8.613644193172456770848011834828, −6.91764756835016110771546797330, −6.28514096937558453864195523927, −4.00126213727776887712711958504, −2.47473631653914824478364614979,
2.07813200094351623362048369923, 4.10105825201638125448174337138, 5.90377330477418653159297661449, 6.56547906066419388604444332290, 8.751251562101132845497298782935, 9.612024742923807442482917825286, 10.17380379886242029895542737338, 11.38012694445929410083328778366, 12.96934189602522066448103425824, 13.82359697871449011233308277260