| L(s) = 1 | − 1.73·3-s + 4i·5-s − 6.92i·7-s + 2.99·9-s − 6.92·11-s − 6.92i·15-s − 18·17-s + 20.7·19-s + 11.9i·21-s − 41.5i·23-s + 9·25-s − 5.19·27-s + 4i·29-s − 48.4i·31-s + 11.9·33-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.800i·5-s − 0.989i·7-s + 0.333·9-s − 0.629·11-s − 0.461i·15-s − 1.05·17-s + 1.09·19-s + 0.571i·21-s − 1.80i·23-s + 0.359·25-s − 0.192·27-s + 0.137i·29-s − 1.56i·31-s + 0.363·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.643449 - 0.643449i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.643449 - 0.643449i\) |
| \(L(2)\) |
|
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 | \( 1 - 4iT - 25T^{2} \) |
| 7 | \( 1 + 6.92iT - 49T^{2} \) |
| 11 | \( 1 + 6.92T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 + 18T + 289T^{2} \) |
| 19 | \( 1 - 20.7T + 361T^{2} \) |
| 23 | \( 1 + 41.5iT - 529T^{2} \) |
| 29 | \( 1 - 4iT - 841T^{2} \) |
| 31 | \( 1 + 48.4iT - 961T^{2} \) |
| 37 | \( 1 + 72iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 18T + 1.68e3T^{2} \) |
| 43 | \( 1 + 62.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 41.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 44iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 62.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 72iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 20.7T + 4.48e3T^{2} \) |
| 71 | \( 1 - 41.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 82T + 5.32e3T^{2} \) |
| 79 | \( 1 - 62.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 131.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 126T + 7.92e3T^{2} \) |
| 97 | \( 1 - 110T + 9.40e3T^{2} \) |
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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
−10.73094107237014905439609927773, −10.39095083551836165124714661943, −9.200118502068570702740929582274, −7.83467195774336091034254854193, −7.04548469267014271353779973728, −6.25609057044065710435807964904, −4.96186869166213398999507704602, −3.89001077513743863548863622670, −2.45595250211152728058045069491, −0.44727877567296724055635898644,
1.46485556768182803787052876897, 3.10915964936445703698215728781, 4.84188599838223589715607340585, 5.33239788539587570502146866863, 6.45622806814619324450059124112, 7.66498121682739745034298865301, 8.713331486732269012787441086133, 9.425228102674133101172910966798, 10.47207689779778097835594594516, 11.62133574083374055303550016168
