| L(s) = 1 | + 1.73·3-s + 2i·5-s − 6.92i·7-s + 2.99·9-s − 6.92·11-s + 2i·13-s + 3.46i·15-s + 10·17-s + 20.7·19-s − 11.9i·21-s − 27.7i·23-s + 21·25-s + 5.19·27-s − 26i·29-s − 6.92i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.400i·5-s − 0.989i·7-s + 0.333·9-s − 0.629·11-s + 0.153i·13-s + 0.230i·15-s + 0.588·17-s + 1.09·19-s − 0.571i·21-s − 1.20i·23-s + 0.839·25-s + 0.192·27-s − 0.896i·29-s − 0.223i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.213672772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.213672772\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73T \) |
| good | 5 | \( 1 - 2iT - 25T^{2} \) |
| 7 | \( 1 + 6.92iT - 49T^{2} \) |
| 11 | \( 1 + 6.92T + 121T^{2} \) |
| 13 | \( 1 - 2iT - 169T^{2} \) |
| 17 | \( 1 - 10T + 289T^{2} \) |
| 19 | \( 1 - 20.7T + 361T^{2} \) |
| 23 | \( 1 + 27.7iT - 529T^{2} \) |
| 29 | \( 1 + 26iT - 841T^{2} \) |
| 31 | \( 1 + 6.92iT - 961T^{2} \) |
| 37 | \( 1 + 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 58T + 1.68e3T^{2} \) |
| 43 | \( 1 - 48.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 69.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 74iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 90.0T + 3.48e3T^{2} \) |
| 61 | \( 1 - 26iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 6.92T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 46T + 5.32e3T^{2} \) |
| 79 | \( 1 + 117. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 48.4T + 6.88e3T^{2} \) |
| 89 | \( 1 + 82T + 7.92e3T^{2} \) |
| 97 | \( 1 - 2T + 9.40e3T^{2} \) |
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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
−10.21811329976046657229011533562, −9.186688048042126597502301912303, −8.195268464366255782245433650483, −7.42877445914674129557775659449, −6.78888280996965155757179153754, −5.51889013897806242161128278352, −4.39067425557841940934984920338, −3.42440478550091800396175655770, −2.40640543679566023511988799430, −0.794244486549894747179730714744,
1.30691959983757672093058543914, 2.67245484847989402229041147747, 3.51850340438285808783819994899, 5.06321940600278503217096529031, 5.51148820447471515505131289671, 6.87150322859514406891215802442, 7.83466722838278243379284497917, 8.526565070847728085696139501427, 9.341463140634068408439714834470, 9.985632794089001454554632744103
