L(s) = 1 | + (−0.974 + 0.222i)2-s + (0.900 − 0.433i)4-s + (−0.0990 − 0.433i)5-s + (−0.781 + 0.623i)8-s + (−0.623 − 0.781i)9-s + (0.193 + 0.400i)10-s + (−0.777 + 0.974i)13-s + (0.623 − 0.781i)16-s + 1.80i·17-s + (0.781 + 0.623i)18-s + (−0.277 − 0.347i)20-s + (0.722 − 0.347i)25-s + (0.541 − 1.12i)26-s + (−0.433 + 0.900i)32-s + (−0.400 − 1.75i)34-s + ⋯ |
L(s) = 1 | + (−0.974 + 0.222i)2-s + (0.900 − 0.433i)4-s + (−0.0990 − 0.433i)5-s + (−0.781 + 0.623i)8-s + (−0.623 − 0.781i)9-s + (0.193 + 0.400i)10-s + (−0.777 + 0.974i)13-s + (0.623 − 0.781i)16-s + 1.80i·17-s + (0.781 + 0.623i)18-s + (−0.277 − 0.347i)20-s + (0.722 − 0.347i)25-s + (0.541 − 1.12i)26-s + (−0.433 + 0.900i)32-s + (−0.400 − 1.75i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5840750918\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5840750918\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.974 - 0.222i)T \) |
| 29 | \( 1 \) |
good | 3 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 5 | \( 1 + (0.0990 + 0.433i)T + (-0.900 + 0.433i)T^{2} \) |
| 7 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 11 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 13 | \( 1 + (0.777 - 0.974i)T + (-0.222 - 0.974i)T^{2} \) |
| 17 | \( 1 - 1.80iT - T^{2} \) |
| 19 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 23 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 31 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 37 | \( 1 + (0.974 - 0.777i)T + (0.222 - 0.974i)T^{2} \) |
| 41 | \( 1 - 1.24iT - T^{2} \) |
| 43 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 47 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 53 | \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + (-0.781 + 1.62i)T + (-0.623 - 0.781i)T^{2} \) |
| 67 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 71 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 73 | \( 1 + (-1.75 - 0.400i)T + (0.900 + 0.433i)T^{2} \) |
| 79 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 83 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 89 | \( 1 + (-0.433 + 0.0990i)T + (0.900 - 0.433i)T^{2} \) |
| 97 | \( 1 + (0.541 + 1.12i)T + (-0.623 + 0.781i)T^{2} \) |
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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
−8.829734248091205946310940416889, −8.398114391014519469470200275753, −7.61704356875073313757185443011, −6.62701414249471753688980251724, −6.27858093730068970362264529830, −5.32237871382586041824772167431, −4.34297543075486433320507045086, −3.28398883045224369517308600235, −2.21137288541667874376231964288, −1.17016847064011474559253113649,
0.51653215604769184355076783554, 2.21087949270746761035978958506, 2.76291917841630634406854080912, 3.60938549436913773905229196575, 5.10486405529229733403104855899, 5.53114744604239880219392343469, 6.87664516708936405667236485105, 7.22069911868347116960200733966, 7.939178991286299186090407198080, 8.682487375725434083964644591817