L(s) = 1 | + (−0.414 − 0.414i)5-s + 3.69i·7-s + (2.93 + 2.93i)11-s + (2.41 − 2.41i)13-s + 2.82·17-s + (−4.46 + 4.46i)19-s − 6.75i·23-s − 4.65i·25-s + (5.24 − 5.24i)29-s + 3.06·31-s + (1.53 − 1.53i)35-s + (6.41 + 6.41i)37-s − 4i·41-s + (0.765 + 0.765i)43-s + 3.06·47-s + ⋯ |
L(s) = 1 | + (−0.185 − 0.185i)5-s + 1.39i·7-s + (0.883 + 0.883i)11-s + (0.669 − 0.669i)13-s + 0.685·17-s + (−1.02 + 1.02i)19-s − 1.40i·23-s − 0.931i·25-s + (0.973 − 0.973i)29-s + 0.549·31-s + (0.258 − 0.258i)35-s + (1.05 + 1.05i)37-s − 0.624i·41-s + (0.116 + 0.116i)43-s + 0.446·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.089748807\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.089748807\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.414 + 0.414i)T + 5iT^{2} \) |
| 7 | \( 1 - 3.69iT - 7T^{2} \) |
| 11 | \( 1 + (-2.93 - 2.93i)T + 11iT^{2} \) |
| 13 | \( 1 + (-2.41 + 2.41i)T - 13iT^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + (4.46 - 4.46i)T - 19iT^{2} \) |
| 23 | \( 1 + 6.75iT - 23T^{2} \) |
| 29 | \( 1 + (-5.24 + 5.24i)T - 29iT^{2} \) |
| 31 | \( 1 - 3.06T + 31T^{2} \) |
| 37 | \( 1 + (-6.41 - 6.41i)T + 37iT^{2} \) |
| 41 | \( 1 + 4iT - 41T^{2} \) |
| 43 | \( 1 + (-0.765 - 0.765i)T + 43iT^{2} \) |
| 47 | \( 1 - 3.06T + 47T^{2} \) |
| 53 | \( 1 + (-3.24 - 3.24i)T + 53iT^{2} \) |
| 59 | \( 1 + (0.765 + 0.765i)T + 59iT^{2} \) |
| 61 | \( 1 + (-0.757 + 0.757i)T - 61iT^{2} \) |
| 67 | \( 1 + (1.39 - 1.39i)T - 67iT^{2} \) |
| 71 | \( 1 - 8.02iT - 71T^{2} \) |
| 73 | \( 1 - 6.48iT - 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 + (9.68 - 9.68i)T - 83iT^{2} \) |
| 89 | \( 1 + 4.82iT - 89T^{2} \) |
| 97 | \( 1 - 5.17T + 97T^{2} \) |
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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
−8.313682721833251211196452118796, −8.056904641115077148452949937927, −6.75249128779444775969631352700, −6.18441601307633725895670897079, −5.64969358922429245077476117983, −4.55042142477088837279257223656, −4.06502330215522688204867066526, −2.83204106152622946562057164408, −2.17490900523370273450167153554, −0.967463228902003845746388201196,
0.76025280047799587773064252079, 1.52665349528921096232533622917, 3.01981250196201068052736523674, 3.76806372847026686050977646805, 4.24367320884997377532407292513, 5.25819728014780679222133476159, 6.27315734818426820640912209247, 6.75505887116379086272774546120, 7.45943715275908357324999816500, 8.140397882085774998948670899993