L(s) = 1 | + (0.575 − 1.29i)2-s + (−1.33 − 1.48i)4-s − 0.311i·5-s + i·7-s + (−2.69 + 0.870i)8-s + (−0.402 − 0.179i)10-s − 3.59·11-s − 6.44·13-s + (1.29 + 0.575i)14-s + (−0.425 + 3.97i)16-s − 2.96i·17-s + 0.684i·19-s + (−0.463 + 0.416i)20-s + (−2.06 + 4.64i)22-s − 7.58·23-s + ⋯ |
L(s) = 1 | + (0.407 − 0.913i)2-s + (−0.668 − 0.743i)4-s − 0.139i·5-s + 0.377i·7-s + (−0.951 + 0.307i)8-s + (−0.127 − 0.0566i)10-s − 1.08·11-s − 1.78·13-s + (0.345 + 0.153i)14-s + (−0.106 + 0.994i)16-s − 0.718i·17-s + 0.157i·19-s + (−0.103 + 0.0930i)20-s + (−0.441 + 0.989i)22-s − 1.58·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.668 - 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.147795 + 0.331521i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.147795 + 0.331521i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.575 + 1.29i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + 0.311iT - 5T^{2} \) |
| 11 | \( 1 + 3.59T + 11T^{2} \) |
| 13 | \( 1 + 6.44T + 13T^{2} \) |
| 17 | \( 1 + 2.96iT - 17T^{2} \) |
| 19 | \( 1 - 0.684iT - 19T^{2} \) |
| 23 | \( 1 + 7.58T + 23T^{2} \) |
| 29 | \( 1 + 9.26iT - 29T^{2} \) |
| 31 | \( 1 - 1.91iT - 31T^{2} \) |
| 37 | \( 1 + 8.40T + 37T^{2} \) |
| 41 | \( 1 + 7.50iT - 41T^{2} \) |
| 43 | \( 1 - 9.30iT - 43T^{2} \) |
| 47 | \( 1 - 9.82T + 47T^{2} \) |
| 53 | \( 1 - 5.43iT - 53T^{2} \) |
| 59 | \( 1 + 2.29T + 59T^{2} \) |
| 61 | \( 1 - 0.486T + 61T^{2} \) |
| 67 | \( 1 - 1.09iT - 67T^{2} \) |
| 71 | \( 1 + 1.76T + 71T^{2} \) |
| 73 | \( 1 - 3.32T + 73T^{2} \) |
| 79 | \( 1 - 1.39iT - 79T^{2} \) |
| 83 | \( 1 + 0.283T + 83T^{2} \) |
| 89 | \( 1 + 8.91iT - 89T^{2} \) |
| 97 | \( 1 - 0.621T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.993155595588284315144244924764, −9.248130549330476135549553984136, −8.196435860505689911710730081412, −7.26622923075764695394367921919, −5.88178236568537734686401419224, −5.11059357351348712918111748469, −4.32403072556945479286109919970, −2.86161514208640678169017413969, −2.17118335378283223458549589691, −0.14786381382789634285743781993,
2.45932761999878677505500660668, 3.66697157783992748468611354385, 4.83373783257520615156168069589, 5.42646156185417258657024243013, 6.64789757452093249766754306069, 7.36986771113401326034380472943, 8.047217294428024488829422378437, 8.991414824758269037461114430680, 10.06231093450915705845537227956, 10.63632024708353233238137087559