L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.707 − 0.707i)3-s + (0.500 − 0.866i)6-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (−0.965 + 0.258i)13-s + (−0.866 − 0.500i)14-s + (−0.5 + 0.866i)16-s + (0.258 + 0.965i)17-s + (−0.965 + 0.258i)18-s + (−0.866 + 0.5i)19-s + 1.00·21-s − 0.999i·24-s + (−0.499 − 0.866i)26-s + (0.707 − 0.707i)27-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.707 − 0.707i)3-s + (0.500 − 0.866i)6-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (−0.965 + 0.258i)13-s + (−0.866 − 0.500i)14-s + (−0.5 + 0.866i)16-s + (0.258 + 0.965i)17-s + (−0.965 + 0.258i)18-s + (−0.866 + 0.5i)19-s + 1.00·21-s − 0.999i·24-s + (−0.499 − 0.866i)26-s + (0.707 − 0.707i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9054942953\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9054942953\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 2 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + 2T + T^{2} \) |
| 73 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
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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
−10.11015407920497134423471998874, −8.713613954801187615977324892021, −8.058327680695988149749650229176, −7.25771320004280098029946632312, −6.32942350231249235149366614891, −6.20943784108573504351208588879, −5.21266842565907864554452232935, −4.43204537611414989455405025752, −2.74422780804718044332332636580, −1.73525798924145876999627885531,
0.70464436383444608646009294590, 2.50623452848331492808740366331, 3.32193752729075552423907321975, 4.28421835543227617220362949836, 4.85139110737766433532986403906, 6.07067229145927456801669299032, 6.93974041419358657816894768104, 7.53172417949998905016521782527, 8.993908124173936070707999165623, 9.749032789423466325775889919518