| L(s) = 1 | + (5.53 + 9.58i)2-s + (136. + 32.8i)3-s + (194. − 337. i)4-s + (312.5 − 541. i)5-s + (439. + 1.48e3i)6-s + (−2.33e3 − 4.05e3i)7-s + 9.97e3·8-s + (1.75e4 + 8.97e3i)9-s + 6.91e3·10-s + (−3.42e4 − 5.93e4i)11-s + (3.76e4 − 3.95e4i)12-s + (2.05e4 − 3.55e4i)13-s + (2.58e4 − 4.48e4i)14-s + (6.04e4 − 6.35e4i)15-s + (−4.44e4 − 7.69e4i)16-s − 3.96e5·17-s + ⋯ |
| L(s) = 1 | + (0.244 + 0.423i)2-s + (0.972 + 0.234i)3-s + (0.380 − 0.658i)4-s + (0.223 − 0.387i)5-s + (0.138 + 0.469i)6-s + (−0.368 − 0.637i)7-s + 0.861·8-s + (0.890 + 0.455i)9-s + 0.218·10-s + (−0.705 − 1.22i)11-s + (0.524 − 0.551i)12-s + (0.199 − 0.344i)13-s + (0.180 − 0.312i)14-s + (0.308 − 0.324i)15-s + (−0.169 − 0.293i)16-s − 1.15·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 + 0.732i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.680 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.06477 - 1.33641i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.06477 - 1.33641i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-136. - 32.8i)T \) |
| 5 | \( 1 + (-312.5 + 541. i)T \) |
| good | 2 | \( 1 + (-5.53 - 9.58i)T + (-256 + 443. i)T^{2} \) |
| 7 | \( 1 + (2.33e3 + 4.05e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (3.42e4 + 5.93e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-2.05e4 + 3.55e4i)T + (-5.30e9 - 9.18e9i)T^{2} \) |
| 17 | \( 1 + 3.96e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.49e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + (-3.97e4 + 6.89e4i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (-1.92e6 - 3.32e6i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 + (-2.21e6 + 3.83e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 - 1.52e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-2.61e5 + 4.53e5i)T + (-1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (-1.36e7 - 2.37e7i)T + (-2.51e14 + 4.35e14i)T^{2} \) |
| 47 | \( 1 + (-6.19e6 - 1.07e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 - 7.97e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + (7.60e7 - 1.31e8i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (9.76e6 + 1.69e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (1.20e8 - 2.09e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + 2.11e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.03e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + (1.51e8 + 2.62e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-1.93e8 - 3.35e8i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 - 2.21e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (8.12e8 + 1.40e9i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
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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
−13.65347681375599167665827024709, −13.23677043655755224425785638655, −10.99938307602546327757977281670, −10.07013173881042954977649169001, −8.714178443269833852433262398588, −7.41633423176483416714755952009, −5.96310505929676114308696826758, −4.43900655581747950065342322635, −2.71386069636870607375979476495, −0.952943584969128876169701435465,
2.04315444526990253482220733425, 2.78929645187024872930570632401, 4.32798442981806363036616932483, 6.67457074849458653695284826580, 7.75148276596572506849012242261, 9.109620732815825886001787034105, 10.39955524951107684191983179146, 11.94128842720801025428434810517, 12.89007801078234502520265734634, 13.73480539826958060003398577875
