| L(s) = 1 | + (1.41 − 0.0909i)2-s + (0.891 − 0.453i)3-s + (1.98 − 0.256i)4-s + (−1.83 + 1.28i)5-s + (1.21 − 0.721i)6-s + (1.46 − 1.46i)7-s + (2.77 − 0.542i)8-s + (0.587 − 0.809i)9-s + (−2.47 + 1.97i)10-s + (−0.716 − 0.986i)11-s + (1.65 − 1.12i)12-s + (0.610 − 0.0966i)13-s + (1.93 − 2.19i)14-s + (−1.05 + 1.97i)15-s + (3.86 − 1.01i)16-s + (−2.23 + 4.38i)17-s + ⋯ |
| L(s) = 1 | + (0.997 − 0.0643i)2-s + (0.514 − 0.262i)3-s + (0.991 − 0.128i)4-s + (−0.819 + 0.572i)5-s + (0.496 − 0.294i)6-s + (0.553 − 0.553i)7-s + (0.981 − 0.191i)8-s + (0.195 − 0.269i)9-s + (−0.781 + 0.624i)10-s + (−0.216 − 0.297i)11-s + (0.476 − 0.325i)12-s + (0.169 − 0.0267i)13-s + (0.516 − 0.587i)14-s + (−0.271 + 0.509i)15-s + (0.967 − 0.254i)16-s + (−0.541 + 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.48073 - 0.316736i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.48073 - 0.316736i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 + 0.0909i)T \) |
| 3 | \( 1 + (-0.891 + 0.453i)T \) |
| 5 | \( 1 + (1.83 - 1.28i)T \) |
| good | 7 | \( 1 + (-1.46 + 1.46i)T - 7iT^{2} \) |
| 11 | \( 1 + (0.716 + 0.986i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.610 + 0.0966i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (2.23 - 4.38i)T + (-9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (-1.11 - 3.43i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (6.81 + 1.07i)T + (21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (-0.953 - 0.309i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (10.1 - 3.28i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.749 + 4.73i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (4.46 + 3.24i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (2.74 + 2.74i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.593 + 1.16i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-2.82 - 5.54i)T + (-31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-10.5 - 7.69i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-7.64 + 5.55i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-7.24 - 3.68i)T + (39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (12.8 + 4.16i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.08 + 6.86i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (2.17 - 6.68i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.19 + 6.27i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (-8.41 - 11.5i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-3.66 + 1.86i)T + (57.0 - 78.4i)T^{2} \) |
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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
−11.86030445273481678699950077969, −10.86115264981607901895138574069, −10.31842507702898949156985003189, −8.475252509782940290450947888026, −7.69451309699510242858336389686, −6.87136460200535635282768498946, −5.69799652290166958026910243405, −4.14613833865527600190888049490, −3.54007431549158958330193956719, −1.96397433219013559733189863403,
2.15299725768047453074349256448, 3.56256842306952747642679405995, 4.64245006243247619945718933574, 5.38236883383984986499366508868, 6.96679743540726234405233402829, 7.88405706383977135008094174204, 8.739269938815768143227867044390, 9.944888123848523068877197731108, 11.45027009464087041149449841335, 11.61083210711503254210169052607
