L(s) = 1 | + (−0.309 − 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.427 + 1.31i)5-s + (−0.309 + 0.951i)6-s + (0.809 − 0.587i)7-s + (0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + 1.38·10-s + (−0.309 − 3.30i)11-s + 0.999·12-s + (−0.381 − 1.17i)13-s + (−0.809 − 0.587i)14-s + (1.11 − 0.812i)15-s + (0.309 − 0.951i)16-s + (2.35 − 7.24i)17-s + ⋯ |
L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.467 − 0.339i)3-s + (−0.404 + 0.293i)4-s + (−0.190 + 0.587i)5-s + (−0.126 + 0.388i)6-s + (0.305 − 0.222i)7-s + (0.286 + 0.207i)8-s + (0.103 + 0.317i)9-s + 0.437·10-s + (−0.0931 − 0.995i)11-s + 0.288·12-s + (−0.105 − 0.326i)13-s + (−0.216 − 0.157i)14-s + (0.288 − 0.209i)15-s + (0.0772 − 0.237i)16-s + (0.570 − 1.75i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.250964 - 0.714157i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.250964 - 0.714157i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 3 | \( 1 + (0.809 + 0.587i)T \) |
| 7 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 + (0.309 + 3.30i)T \) |
good | 5 | \( 1 + (0.427 - 1.31i)T + (-4.04 - 2.93i)T^{2} \) |
| 13 | \( 1 + (0.381 + 1.17i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.35 + 7.24i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (3.30 + 2.40i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + 9.32T + 23T^{2} \) |
| 29 | \( 1 + (-1 + 0.726i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.336 - 1.03i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.97 + 5.06i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (9.59 + 6.96i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 5.23T + 43T^{2} \) |
| 47 | \( 1 + (8.23 + 5.98i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-2.76 - 8.50i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-7.23 + 5.25i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (2.85 - 8.78i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + 1.23T + 67T^{2} \) |
| 71 | \( 1 + (-0.472 + 1.45i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-6.47 + 4.70i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (2.38 + 7.33i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-4.70 + 14.4i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 5.14T + 89T^{2} \) |
| 97 | \( 1 + (-3.47 - 10.6i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.77510965548990309119660977936, −10.11835628363836559146574155700, −8.958705185671186025812355851062, −7.931894702525411979018098363031, −7.15247380164770849142465848160, −5.97117666017321595232030540762, −4.87194595095778443472059548526, −3.53505822108427828487962040709, −2.37534112523506956917378762663, −0.54236191123585948230760585494,
1.71994509486623232268839802055, 4.02556044693945944124665351080, 4.71905236663916882994324968400, 5.86265304610921556627310879370, 6.60946481345773416199663192588, 8.076937112992542481300009992854, 8.352205833695885609840542014302, 9.800366238764075969424072279156, 10.15756854108297848718338722931, 11.39731926223385019909563611951