| L(s) = 1 | + (4 + 6.92i)2-s + (−15.9 − 27.6i)3-s + (−31.9 + 55.4i)4-s − 54.4·5-s + (127. − 220. i)6-s + (556. − 963. i)7-s − 511.·8-s + (585. − 1.01e3i)9-s + (−217. − 377. i)10-s + (−3.56e3 − 6.17e3i)11-s + 2.03e3·12-s + (7.56e3 + 2.34e3i)13-s + 8.90e3·14-s + (867. + 1.50e3i)15-s + (−2.04e3 − 3.54e3i)16-s + (1.02e4 − 1.77e4i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.340 − 0.590i)3-s + (−0.249 + 0.433i)4-s − 0.194·5-s + (0.240 − 0.417i)6-s + (0.613 − 1.06i)7-s − 0.353·8-s + (0.267 − 0.463i)9-s + (−0.0688 − 0.119i)10-s + (−0.807 − 1.39i)11-s + 0.340·12-s + (0.955 + 0.296i)13-s + 0.867·14-s + (0.0663 + 0.114i)15-s + (−0.125 − 0.216i)16-s + (0.506 − 0.876i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.32876 - 0.796255i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32876 - 0.796255i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4 - 6.92i)T \) |
| 13 | \( 1 + (-7.56e3 - 2.34e3i)T \) |
| good | 3 | \( 1 + (15.9 + 27.6i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + 54.4T + 7.81e4T^{2} \) |
| 7 | \( 1 + (-556. + 963. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (3.56e3 + 6.17e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 17 | \( 1 + (-1.02e4 + 1.77e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.40e4 - 2.43e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (1.69e4 + 2.93e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-8.77e4 - 1.51e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 1.56e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + (998. + 1.72e3i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (1.11e5 + 1.93e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-2.68e5 + 4.65e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 - 5.42e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.85e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-6.65e5 + 1.15e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.43e6 - 2.48e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.41e6 - 2.45e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-8.00e5 + 1.38e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 - 1.39e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.33e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.37e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (3.52e6 + 6.10e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (4.25e6 - 7.37e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−15.84032261032378979759255502373, −14.17034724646112985847384806431, −13.41791173636955681392993540769, −11.96292496787501535742903930285, −10.63579854752849494014744812124, −8.412601132366499250411333218446, −7.20767196296221803248795093025, −5.78443546173782474486460611623, −3.83752993980241522668129601175, −0.77624088404093377737204112699,
2.10824556057425073623084577750, 4.36270168228763664302675118250, 5.61392653010655724868642819762, 8.082845610661017241180821525442, 9.832907861596866597401421165562, 10.94936682865697185767004653392, 12.14591695924005880568788672485, 13.35578815406339104687970231013, 15.17594893357944776738699853944, 15.61343049491279625869924254534
