L(s) = 1 | + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s + (0.951 + 1.30i)13-s + (−0.809 + 0.587i)16-s + (−0.587 − 0.190i)17-s + i·18-s − 1.61·26-s + (−0.190 − 0.587i)29-s − i·32-s + (0.5 − 0.363i)34-s + (−0.809 − 0.587i)36-s + (0.363 + 0.5i)37-s + (1.30 − 0.951i)41-s + ⋯ |
L(s) = 1 | + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s + (0.951 + 1.30i)13-s + (−0.809 + 0.587i)16-s + (−0.587 − 0.190i)17-s + i·18-s − 1.61·26-s + (−0.190 − 0.587i)29-s − i·32-s + (0.5 − 0.363i)34-s + (−0.809 − 0.587i)36-s + (0.363 + 0.5i)37-s + (1.30 − 0.951i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9596693531\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9596693531\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.587 - 0.809i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \) |
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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
−9.078384237919109956829406438384, −8.567948820875594678599351710943, −7.58830408453701612413111884670, −6.85004440466613461652368538365, −6.39383604982125255962776850392, −5.52510234716843048877038324838, −4.39263511120083179412647862269, −3.91606213651111719508366354945, −2.19477488653037964372345342282, −1.10799447974058672047951640002,
1.05011273037368801703507147097, 2.13760148543456719917480459789, 3.16087552154813809890806011833, 4.02219669800612266713196779399, 4.85150707528879494676363011036, 5.89104972021015955691712935521, 6.94747097660788011730607683384, 7.74921776409781324221501610650, 8.270903056745153199843245102684, 9.073109017940789330124878489957