| L(s) = 1 | + (−0.766 − 0.642i)3-s + (−0.800 + 0.291i)5-s + (−1.70 + 0.983i)7-s + (0.173 + 0.984i)9-s + (−0.742 − 0.428i)11-s + (2.79 + 3.33i)13-s + (0.800 + 0.291i)15-s + (1.26 − 7.14i)17-s + (0.199 − 4.35i)19-s + (1.93 + 0.341i)21-s + (0.620 − 1.70i)23-s + (−3.27 + 2.74i)25-s + (0.500 − 0.866i)27-s + (9.99 − 1.76i)29-s + (−1.88 − 3.25i)31-s + ⋯ |
| L(s) = 1 | + (−0.442 − 0.371i)3-s + (−0.357 + 0.130i)5-s + (−0.644 + 0.371i)7-s + (0.0578 + 0.328i)9-s + (−0.223 − 0.129i)11-s + (0.776 + 0.925i)13-s + (0.206 + 0.0752i)15-s + (0.305 − 1.73i)17-s + (0.0456 − 0.998i)19-s + (0.422 + 0.0745i)21-s + (0.129 − 0.355i)23-s + (−0.654 + 0.549i)25-s + (0.0962 − 0.166i)27-s + (1.85 − 0.327i)29-s + (−0.337 − 0.584i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.687113 - 0.618100i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.687113 - 0.618100i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.766 + 0.642i)T \) |
| 19 | \( 1 + (-0.199 + 4.35i)T \) |
| good | 5 | \( 1 + (0.800 - 0.291i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (1.70 - 0.983i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.742 + 0.428i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.79 - 3.33i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.26 + 7.14i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-0.620 + 1.70i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-9.99 + 1.76i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (1.88 + 3.25i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 8.42iT - 37T^{2} \) |
| 41 | \( 1 + (0.985 - 1.17i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (3.98 + 10.9i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (7.46 - 1.31i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-1.86 + 5.11i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.02 + 5.83i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-2.62 - 0.955i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.41 - 8.04i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-11.8 + 4.30i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-4.45 - 3.73i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (3.58 + 3.01i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-11.1 + 6.44i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (6.05 + 7.22i)T + (-15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (7.66 + 1.35i)T + (91.1 + 33.1i)T^{2} \) |
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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
−9.799206153958535159247097626973, −9.131142252492374427014393107197, −8.205218519147352961661605592190, −7.10204754100457962700738643254, −6.63446327142902071703957708989, −5.58274697442588578206438783566, −4.65393914255315885910810388603, −3.41158378865311999503254874715, −2.31154488881688255084508333333, −0.52065191789826119525756846270,
1.26234921495145204647325244141, 3.22219679348613531033916775148, 3.88825728763229181934696019243, 5.01490982984425796481948406540, 6.06888110507270350865572098237, 6.61282088469448289110422593524, 8.115182373053377868337410524369, 8.286853189638245719335579386061, 9.788892239955219806904744707343, 10.26939123583482334380168214443
