| L(s) = 1 | − 3.46i·5-s + 10i·7-s + 13.8·11-s + 6.92i·13-s − 30·17-s − 6.92·19-s − 12i·23-s + 13.0·25-s + 51.9i·29-s + 14i·31-s + 34.6·35-s + 55.4i·37-s + 6·41-s + 62.3·43-s + 84i·47-s + ⋯ |
| L(s) = 1 | − 0.692i·5-s + 1.42i·7-s + 1.25·11-s + 0.532i·13-s − 1.76·17-s − 0.364·19-s − 0.521i·23-s + 0.520·25-s + 1.79i·29-s + 0.451i·31-s + 0.989·35-s + 1.49i·37-s + 0.146·41-s + 1.45·43-s + 1.78i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.541720946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.541720946\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 3.46iT - 25T^{2} \) |
| 7 | \( 1 - 10iT - 49T^{2} \) |
| 11 | \( 1 - 13.8T + 121T^{2} \) |
| 13 | \( 1 - 6.92iT - 169T^{2} \) |
| 17 | \( 1 + 30T + 289T^{2} \) |
| 19 | \( 1 + 6.92T + 361T^{2} \) |
| 23 | \( 1 + 12iT - 529T^{2} \) |
| 29 | \( 1 - 51.9iT - 841T^{2} \) |
| 31 | \( 1 - 14iT - 961T^{2} \) |
| 37 | \( 1 - 55.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 62.3T + 1.84e3T^{2} \) |
| 47 | \( 1 - 84iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 17.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 62.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 96.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 48.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 60iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 86T + 5.32e3T^{2} \) |
| 79 | \( 1 - 38iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 13.8T + 6.88e3T^{2} \) |
| 89 | \( 1 + 78T + 7.92e3T^{2} \) |
| 97 | \( 1 - 62T + 9.40e3T^{2} \) |
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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
−10.90224244428363022207449041198, −9.453411902735921273701493323607, −8.859263750765588976718936343055, −8.532101527460893961976831888999, −6.86218345965042272201075070055, −6.24655490795604291462452845787, −5.03844751664607348139678014881, −4.24792991917049444724925928247, −2.69489393388727772336867656866, −1.47183051499872596619843961828,
0.61283480987577486908157251457, 2.27809206313197820495624340122, 3.81276735275334532379363049809, 4.31880862200366076178991506554, 5.96676204210569846680043446906, 6.86905918201796305504872498717, 7.38893709189746923848017209327, 8.606816162038943483748287018081, 9.552019412062116224604496767825, 10.49028524096563118541734104410
