| L(s) = 1 | + 2.23i·5-s − 8.75i·7-s + 16.6i·11-s − 4.06·13-s − 6.82i·17-s − 24.2i·19-s + (−0.407 + 22.9i)23-s − 5.00·25-s − 3.59·29-s + 18.0·31-s + 19.5·35-s − 2.42i·37-s + 23.8·41-s + 15.8i·43-s − 24.3·47-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s − 1.25i·7-s + 1.51i·11-s − 0.312·13-s − 0.401i·17-s − 1.27i·19-s + (−0.0177 + 0.999i)23-s − 0.200·25-s − 0.123·29-s + 0.583·31-s + 0.559·35-s − 0.0656i·37-s + 0.580·41-s + 0.368i·43-s − 0.518·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0177i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.888344905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.888344905\) |
| \(L(2)\) |
|
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 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (0.407 - 22.9i)T \) |
| good | 7 | \( 1 + 8.75iT - 49T^{2} \) |
| 11 | \( 1 - 16.6iT - 121T^{2} \) |
| 13 | \( 1 + 4.06T + 169T^{2} \) |
| 17 | \( 1 + 6.82iT - 289T^{2} \) |
| 19 | \( 1 + 24.2iT - 361T^{2} \) |
| 29 | \( 1 + 3.59T + 841T^{2} \) |
| 31 | \( 1 - 18.0T + 961T^{2} \) |
| 37 | \( 1 + 2.42iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 23.8T + 1.68e3T^{2} \) |
| 43 | \( 1 - 15.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 24.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 37.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 66.7T + 3.48e3T^{2} \) |
| 61 | \( 1 - 17.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 34.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 89.2T + 5.04e3T^{2} \) |
| 73 | \( 1 - 19.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 20.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 93.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 94.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 95.4iT - 9.40e3T^{2} \) |
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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
−7.993590530691121172809225068471, −7.27282689662468140570619924118, −7.09158245937989188159712784939, −6.22937301761337599524709504763, −5.00986676863668104712131471870, −4.54619691781401915775777554096, −3.69525036705054382145124591536, −2.74117267661236660699083123250, −1.78732709711818869928530561314, −0.62367317611488546227324246017,
0.63004205951296595849818785266, 1.81738753838834391932158033747, 2.75309126745690622115265190042, 3.56170419434426377954603122628, 4.54150661639835044361770831483, 5.46407819044308412158050138107, 5.94806653903615910342748998095, 6.53281668569734078044357598050, 7.83689339978217645649304383806, 8.305776875037060546162130671536
