L(s) = 1 | + (0.5 + 0.866i)2-s + 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)16-s + (0.499 − 0.866i)18-s + 0.999·22-s + (0.5 + 0.866i)23-s − 29-s + (0.5 + 0.866i)37-s − 43-s + (−0.499 + 0.866i)46-s + (−1 + 1.73i)53-s + (−0.5 − 0.866i)58-s + 64-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)16-s + (0.499 − 0.866i)18-s + 0.999·22-s + (0.5 + 0.866i)23-s − 29-s + (0.5 + 0.866i)37-s − 43-s + (−0.499 + 0.866i)46-s + (−1 + 1.73i)53-s + (−0.5 − 0.866i)58-s + 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.500352219\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.500352219\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 3 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.818267397437804726328765840831, −9.095461305496279478817919965895, −8.212893250609105686544405094365, −7.33726212827740936246099960690, −6.46455745812204634109356258784, −5.93067135229850284971520874692, −5.12898918028780398710565239715, −4.00807912681082987194580511139, −3.09386995085401625795016973948, −1.36993470125684026314071655134,
1.75470996882189676242365278825, 2.57891524588464892588207153204, 3.66561247166956056647323094004, 4.58160091886499624170858388547, 5.30824885253189725655032890565, 6.59593430063734229162133716292, 7.44912845942094310470123251206, 8.183465245407622570640834812019, 9.189403386069815823512362787838, 10.11273830344097394194785571571