| L(s) = 1 | + 1.73i·3-s − 1.01i·5-s + (2.24 + 6.63i)7-s − 2.99·9-s − 10.2·11-s + 8.95i·13-s + 1.75·15-s + 30.4i·17-s − 16.1i·19-s + (−11.4 + 3.88i)21-s + 6.72·23-s + 23.9·25-s − 5.19i·27-s − 30·29-s − 50.1i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.202i·5-s + (0.320 + 0.947i)7-s − 0.333·9-s − 0.931·11-s + 0.689i·13-s + 0.117·15-s + 1.78i·17-s − 0.849i·19-s + (−0.546 + 0.184i)21-s + 0.292·23-s + 0.958·25-s − 0.192i·27-s − 1.03·29-s − 1.61i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.320i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6127640401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6127640401\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 7 | \( 1 + (-2.24 - 6.63i)T \) |
| good | 5 | \( 1 + 1.01iT - 25T^{2} \) |
| 11 | \( 1 + 10.2T + 121T^{2} \) |
| 13 | \( 1 - 8.95iT - 169T^{2} \) |
| 17 | \( 1 - 30.4iT - 289T^{2} \) |
| 19 | \( 1 + 16.1iT - 361T^{2} \) |
| 23 | \( 1 - 6.72T + 529T^{2} \) |
| 29 | \( 1 + 30T + 841T^{2} \) |
| 31 | \( 1 + 50.1iT - 961T^{2} \) |
| 37 | \( 1 + 30.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 7.10iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 74.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 58.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 70.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 0.492iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 2.86iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 27.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 50.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + 70.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 133.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 104. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 144. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 100. iT - 9.40e3T^{2} \) |
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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
−9.820727178159810356464282382131, −8.938971090053323846312605819191, −8.498140199347268793946876541546, −7.58712633973860014808918497641, −6.41865182798310497144751210601, −5.58663539995694140345922197731, −4.87276500426580781272480595782, −3.94548569413904397002695092774, −2.76460589225667875484154077062, −1.76772055867422893577125506541,
0.17160348618925721664823209766, 1.35938692557850369150529957440, 2.71811408849683900423507274506, 3.54924719395106220760448034082, 4.97704024230307590584101326808, 5.43373342390183975313227739943, 6.92855996784070789005796299510, 7.16353274591773058086074166474, 8.070029880441235853973711794907, 8.786941510568536388324853726716
