| L(s) = 1 | + (−91.8 + 159. i)3-s + (−609. − 1.05e3i)5-s + (2.97e3 + 5.61e3i)7-s + (−7.02e3 − 1.21e4i)9-s + (2.07e3 − 3.59e3i)11-s − 1.07e5·13-s + 2.23e5·15-s + (7.65e4 − 1.32e5i)17-s + (−2.64e4 − 4.57e4i)19-s + (−1.16e6 − 4.15e4i)21-s + (−7.05e5 − 1.22e6i)23-s + (2.32e5 − 4.03e5i)25-s − 1.03e6·27-s + 1.37e6·29-s + (3.33e6 − 5.76e6i)31-s + ⋯ |
| L(s) = 1 | + (−0.654 + 1.13i)3-s + (−0.436 − 0.755i)5-s + (0.468 + 0.883i)7-s + (−0.356 − 0.617i)9-s + (0.0427 − 0.0741i)11-s − 1.04·13-s + 1.14·15-s + (0.222 − 0.384i)17-s + (−0.0464 − 0.0805i)19-s + (−1.30 − 0.0465i)21-s + (−0.525 − 0.910i)23-s + (0.119 − 0.206i)25-s − 0.375·27-s + 0.360·29-s + (0.647 − 1.12i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.215481662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.215481662\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.97e3 - 5.61e3i)T \) |
| good | 3 | \( 1 + (91.8 - 159. i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (609. + 1.05e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (-2.07e3 + 3.59e3i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + 1.07e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + (-7.65e4 + 1.32e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (2.64e4 + 4.57e4i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (7.05e5 + 1.22e6i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 - 1.37e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-3.33e6 + 5.76e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (3.61e6 + 6.25e6i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 + 1.18e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.34e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-1.70e7 - 2.95e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (3.14e7 - 5.44e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (4.44e7 - 7.69e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-7.29e7 - 1.26e8i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-5.27e6 + 9.12e6i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 - 2.87e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (-1.91e8 + 3.32e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-2.57e8 - 4.46e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + 1.77e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (2.14e8 + 3.70e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 - 1.36e9T + 7.60e17T^{2} \) |
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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
−11.93300935103342250197607329331, −10.88852365915001770853759737831, −9.803326776854958775073386378552, −8.886242542499537459394726728325, −7.75111816709574301876012183726, −5.93992639791517025757119131145, −4.91066619597936103532186922710, −4.28800843540482330888895985640, −2.46083584700405267644694707333, −0.53843780160642971634749204439,
0.70276762832331480562583672375, 1.90298692303580942847857487375, 3.55481385591959272475957288631, 5.08786982274312180360648719794, 6.56742654267071216579025541336, 7.24994193361880474828738069005, 8.038270648193210379950851050306, 9.923450991850232532526385048830, 10.95169898007505377521655086784, 11.81528107395901713783887407035
