| L(s) = 1 | − 5.48i·7-s − 9.17i·11-s − 11.4i·13-s − 16.9·17-s + 26.9·19-s − 4.93·23-s + 20.5i·29-s − 20.9·31-s − 62.4i·37-s − 40.9i·41-s + 1.02i·43-s + 86.2·47-s + 18.8·49-s − 96.0·53-s − 112. i·59-s + ⋯ |
| L(s) = 1 | − 0.783i·7-s − 0.833i·11-s − 0.883i·13-s − 0.998·17-s + 1.41·19-s − 0.214·23-s + 0.707i·29-s − 0.676·31-s − 1.68i·37-s − 0.998i·41-s + 0.0238i·43-s + 1.83·47-s + 0.385·49-s − 1.81·53-s − 1.90i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.037612873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.037612873\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 5.48iT - 49T^{2} \) |
| 11 | \( 1 + 9.17iT - 121T^{2} \) |
| 13 | \( 1 + 11.4iT - 169T^{2} \) |
| 17 | \( 1 + 16.9T + 289T^{2} \) |
| 19 | \( 1 - 26.9T + 361T^{2} \) |
| 23 | \( 1 + 4.93T + 529T^{2} \) |
| 29 | \( 1 - 20.5iT - 841T^{2} \) |
| 31 | \( 1 + 20.9T + 961T^{2} \) |
| 37 | \( 1 + 62.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 40.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.02iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 86.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 96.0T + 2.80e3T^{2} \) |
| 59 | \( 1 + 112. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 66.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 76iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 24.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 18.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 106.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 45.1T + 6.88e3T^{2} \) |
| 89 | \( 1 - 115. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 87.0iT - 9.40e3T^{2} \) |
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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
−7.84768801215547529888443685273, −7.42288345393879131489726696769, −6.60216161440733942923222000651, −5.67263850306628788916822425784, −5.13061429982365815839201086442, −3.98482932439845667572321177398, −3.42117034535307123823472185259, −2.40846502192516552927908731190, −1.11701706913567335210857273271, −0.23502285674115102396043897540,
1.39905142018761848014937078673, 2.27348697575540079338853001346, 3.13153721217907828931066330570, 4.30565318846886816978171618336, 4.83463323030238594359939771181, 5.80190022733014085668215777743, 6.48371985163767664217769624907, 7.28299903413175121975849540022, 7.924762292228829895998545736458, 8.926054147059210908237290092782
