L(s) = 1 | + (0.979 + 1.01i)2-s + (−0.582 + 0.582i)3-s + (−0.0796 + 1.99i)4-s + (1.46 − 1.68i)5-s + (−1.16 − 0.0231i)6-s + (−0.972 − 0.972i)7-s + (−2.11 + 1.87i)8-s + 2.32i·9-s + (3.15 − 0.157i)10-s + 4.73i·11-s + (−1.11 − 1.21i)12-s + (4.00 + 4.00i)13-s + (0.0387 − 1.94i)14-s + (0.128 + 1.83i)15-s + (−3.98 − 0.318i)16-s + (1.22 − 1.22i)17-s + ⋯ |
L(s) = 1 | + (0.692 + 0.721i)2-s + (−0.336 + 0.336i)3-s + (−0.0398 + 0.999i)4-s + (0.656 − 0.754i)5-s + (−0.475 − 0.00947i)6-s + (−0.367 − 0.367i)7-s + (−0.748 + 0.663i)8-s + 0.773i·9-s + (0.998 − 0.0496i)10-s + 1.42i·11-s + (−0.322 − 0.349i)12-s + (1.11 + 1.11i)13-s + (0.0103 − 0.519i)14-s + (0.0330 + 0.474i)15-s + (−0.996 − 0.0796i)16-s + (0.297 − 0.297i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.10376 + 1.43816i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.10376 + 1.43816i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.979 - 1.01i)T \) |
| 5 | \( 1 + (-1.46 + 1.68i)T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + (0.582 - 0.582i)T - 3iT^{2} \) |
| 7 | \( 1 + (0.972 + 0.972i)T + 7iT^{2} \) |
| 11 | \( 1 - 4.73iT - 11T^{2} \) |
| 13 | \( 1 + (-4.00 - 4.00i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.22 + 1.22i)T - 17iT^{2} \) |
| 23 | \( 1 + (-1.83 + 1.83i)T - 23iT^{2} \) |
| 29 | \( 1 + 6.49iT - 29T^{2} \) |
| 31 | \( 1 - 1.38iT - 31T^{2} \) |
| 37 | \( 1 + (-3.47 + 3.47i)T - 37iT^{2} \) |
| 41 | \( 1 + 6.68T + 41T^{2} \) |
| 43 | \( 1 + (-4.72 + 4.72i)T - 43iT^{2} \) |
| 47 | \( 1 + (-6.12 - 6.12i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3.65 - 3.65i)T + 53iT^{2} \) |
| 59 | \( 1 - 0.289T + 59T^{2} \) |
| 61 | \( 1 + 4.63T + 61T^{2} \) |
| 67 | \( 1 + (3.20 + 3.20i)T + 67iT^{2} \) |
| 71 | \( 1 + 14.8iT - 71T^{2} \) |
| 73 | \( 1 + (-6.01 - 6.01i)T + 73iT^{2} \) |
| 79 | \( 1 + 5.24T + 79T^{2} \) |
| 83 | \( 1 + (-4.42 + 4.42i)T - 83iT^{2} \) |
| 89 | \( 1 - 8.67iT - 89T^{2} \) |
| 97 | \( 1 + (3.14 - 3.14i)T - 97iT^{2} \) |
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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
−11.87460140330681301898198523192, −10.73300375741904768981425829913, −9.671601480958694221197772341971, −8.850776925189973048540143974953, −7.71108925403918165118701634232, −6.69725187860071518155855657122, −5.76954593683227747703041109053, −4.71863284363260243148835630229, −4.11239192123751600550281775170, −2.14866139711485868506095888771,
1.13308096658030050304865081930, 2.97489203355246319682965505161, 3.56110152591114002938407737456, 5.63056115809047436605061868616, 5.92992755004111535669958397803, 6.82681488238025445710134366086, 8.538878984333503454439902644754, 9.481312327697730054560451023163, 10.55869599100620932025989601266, 11.06026897067105116340678718907