| L(s) = 1 | − 1.41·2-s + 4.18i·3-s + 2.00·4-s + 3.16i·5-s − 5.91i·6-s − 2.82·8-s − 8.48·9-s − 4.47i·10-s − 13.2·11-s + 8.36i·12-s + 5.49i·13-s − 13.2·15-s + 4.00·16-s + 13.5i·17-s + 12.0·18-s − 0.717i·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.39i·3-s + 0.500·4-s + 0.633i·5-s − 0.985i·6-s − 0.353·8-s − 0.942·9-s − 0.447i·10-s − 1.20·11-s + 0.696i·12-s + 0.422i·13-s − 0.882·15-s + 0.250·16-s + 0.797i·17-s + 0.666·18-s − 0.0377i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.283431 + 0.760226i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.283431 + 0.760226i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 4.18iT - 9T^{2} \) |
| 5 | \( 1 - 3.16iT - 25T^{2} \) |
| 11 | \( 1 + 13.2T + 121T^{2} \) |
| 13 | \( 1 - 5.49iT - 169T^{2} \) |
| 17 | \( 1 - 13.5iT - 289T^{2} \) |
| 19 | \( 1 + 0.717iT - 361T^{2} \) |
| 23 | \( 1 + 2.27T + 529T^{2} \) |
| 29 | \( 1 - 20.4T + 841T^{2} \) |
| 31 | \( 1 + 24.6iT - 961T^{2} \) |
| 37 | \( 1 - 64.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 21.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 6.48T + 1.84e3T^{2} \) |
| 47 | \( 1 - 47.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 22.0T + 2.80e3T^{2} \) |
| 59 | \( 1 - 83.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 66.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 92.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 48.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + 130. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 76.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 107. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 167. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 25.5iT - 9.40e3T^{2} \) |
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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
−14.55216752106181677944470669233, −13.01503500970151621184950063367, −11.42681650149681774218172742712, −10.56156282321112357704684007018, −9.984984765823436223573531734614, −8.838948530614137720132247608758, −7.59466875598568499696646299045, −6.01535387850148020994597155367, −4.45550139028975736656622976064, −2.84823006573045957364810800657,
0.78719332633456957535995776613, 2.53023533207106782689177136474, 5.26603376121205831469878018386, 6.72674837061930701824794082812, 7.76572456155644459538457214736, 8.520707584269449788381752231035, 9.936923525599836007299780831419, 11.22872690031717626055166159124, 12.42329586427525741592609811500, 12.98186543381486044848172643785
