L(s) = 1 | − i·2-s + (0.707 + 0.707i)3-s − 4-s + (1.41 + 1.41i)5-s + (0.707 − 0.707i)6-s + i·8-s + 1.00i·9-s + (1.41 − 1.41i)10-s + (−2.82 + 2.82i)11-s + (−0.707 − 0.707i)12-s + 2·13-s + 2.00i·15-s + 16-s + 1.00·18-s − 4i·19-s + (−1.41 − 1.41i)20-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (0.632 + 0.632i)5-s + (0.288 − 0.288i)6-s + 0.353i·8-s + 0.333i·9-s + (0.447 − 0.447i)10-s + (−0.852 + 0.852i)11-s + (−0.204 − 0.204i)12-s + 0.554·13-s + 0.516i·15-s + 0.250·16-s + 0.235·18-s − 0.917i·19-s + (−0.316 − 0.316i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.875569103\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.875569103\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 17 | \( 1 \) |
good | 5 | \( 1 + (-1.41 - 1.41i)T + 5iT^{2} \) |
| 7 | \( 1 - 7iT^{2} \) |
| 11 | \( 1 + (2.82 - 2.82i)T - 11iT^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 23iT^{2} \) |
| 29 | \( 1 + (-7.07 - 7.07i)T + 29iT^{2} \) |
| 31 | \( 1 + (-5.65 - 5.65i)T + 31iT^{2} \) |
| 37 | \( 1 + (1.41 + 1.41i)T + 37iT^{2} \) |
| 41 | \( 1 + (7.07 - 7.07i)T - 41iT^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 + (-7.07 + 7.07i)T - 61iT^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 71iT^{2} \) |
| 73 | \( 1 + (7.07 + 7.07i)T + 73iT^{2} \) |
| 79 | \( 1 + (5.65 - 5.65i)T - 79iT^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + (-9.89 - 9.89i)T + 97iT^{2} \) |
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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
−9.627569212591744685361818828340, −8.779513316601854980470917439544, −8.088453963684844770298565225182, −7.01744557893090597620161380983, −6.24323155993572064555889166174, −5.00037074411665176561237703695, −4.52141574697682164516372558356, −3.07830398946111315135958691862, −2.71123168396102281928516274806, −1.47636988186842666570485247348,
0.69169905610264995419003434904, 2.02802793126179874867724678641, 3.25782635837158200272772839711, 4.31154389756146238155898685732, 5.46395910157928627437454571796, 5.88669135276707711730224983634, 6.78065634721167287798466715701, 7.78727571047300743346359690419, 8.435123725518845428766956404192, 8.825010810829502556297122199962