L(s) = 1 | + (−0.707 + 0.707i)5-s − i·7-s − 1.41i·11-s − i·13-s + 19-s + 1.41·23-s − 1.00i·25-s + 1.41i·29-s + (0.707 + 0.707i)35-s + i·37-s − 1.41i·41-s − 1.41·53-s + (1.00 + 1.00i)55-s − 61-s + (0.707 + 0.707i)65-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)5-s − i·7-s − 1.41i·11-s − i·13-s + 19-s + 1.41·23-s − 1.00i·25-s + 1.41i·29-s + (0.707 + 0.707i)35-s + i·37-s − 1.41i·41-s − 1.41·53-s + (1.00 + 1.00i)55-s − 61-s + (0.707 + 0.707i)65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9060680262\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9060680262\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 + iT - T^{2} \) |
| 11 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 + iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 - 1.41T + T^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - iT - T^{2} \) |
| 41 | \( 1 + 1.41iT - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + 1.41T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + iT - T^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - iT - T^{2} \) |
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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
−10.26112359208267808042379830940, −9.070329486881567158095827032562, −8.200163970668011993192511459432, −7.44975303052875230523096987207, −6.82041338045018042892834334766, −5.74051901653664801545713939703, −4.73583662097147993020629515256, −3.32993078421676406979888763454, −3.20423035322691142272678170533, −0.927855472286996525520908739439,
1.62372811312655153054184665337, 2.87581608948235562614761370286, 4.26603112404279290351173036892, 4.85267865907708623276420154481, 5.83546181186525117480088096184, 7.02236779013932368493349869111, 7.65392256095812764376169841466, 8.668181637061568833567524689654, 9.321798870125563161094676160369, 9.877369600146137111987135667750