L(s) = 1 | + (−0.522 + 0.904i)2-s + (0.392 + 1.68i)3-s + (0.454 + 0.787i)4-s + (−0.5 − 0.866i)5-s + (−1.73 − 0.525i)6-s + (0.5 − 0.866i)7-s − 3.03·8-s + (−2.69 + 1.32i)9-s + 1.04·10-s + (−2.85 + 4.94i)11-s + (−1.15 + 1.07i)12-s + (1.04 + 1.81i)13-s + (0.522 + 0.904i)14-s + (1.26 − 1.18i)15-s + (0.676 − 1.17i)16-s + 3.21·17-s + ⋯ |
L(s) = 1 | + (−0.369 + 0.639i)2-s + (0.226 + 0.973i)3-s + (0.227 + 0.393i)4-s + (−0.223 − 0.387i)5-s + (−0.706 − 0.214i)6-s + (0.188 − 0.327i)7-s − 1.07·8-s + (−0.897 + 0.441i)9-s + 0.330·10-s + (−0.860 + 1.49i)11-s + (−0.332 + 0.310i)12-s + (0.290 + 0.502i)13-s + (0.139 + 0.241i)14-s + (0.326 − 0.305i)15-s + (0.169 − 0.292i)16-s + 0.779·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.238i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.119342 + 0.985775i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.119342 + 0.985775i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.392 - 1.68i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 2 | \( 1 + (0.522 - 0.904i)T + (-1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (2.85 - 4.94i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.04 - 1.81i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 3.21T + 17T^{2} \) |
| 19 | \( 1 + 1.22T + 19T^{2} \) |
| 23 | \( 1 + (1.83 + 3.17i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.588 + 1.02i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.29 - 5.70i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 9.74T + 37T^{2} \) |
| 41 | \( 1 + (-5.12 - 8.87i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.76 + 8.25i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.46 - 5.99i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 4.52T + 53T^{2} \) |
| 59 | \( 1 + (5.91 + 10.2i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.46 - 7.74i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.96 + 5.13i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 - 5.49T + 73T^{2} \) |
| 79 | \( 1 + (6.68 - 11.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.74 - 3.02i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 + (1.04 - 1.81i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.08838616498891063970326822871, −11.02270121987299346921161662334, −10.01547709136029692442313272088, −9.240241401372608523432341666089, −8.154355976696431090770750863023, −7.63161570032433088689723385613, −6.33972706496209583760410641211, −4.97047794359553488275284220634, −4.06144515075988093172036505467, −2.60126898871472014201299932480,
0.77817113777242596726804053584, 2.42827566217118307690145466759, 3.29768806550002899434182306883, 5.69893617948405878453467748220, 6.11869015757645816273983011809, 7.63752845208306950567720509998, 8.307707463510561327274512795138, 9.370075548584898448839504603579, 10.56150283548793337971683459517, 11.24617133931042633227537079250