L(s) = 1 | + (1 − i)5-s + i·13-s − i·25-s + 2·29-s + (1 + i)37-s + (1 − i)41-s − i·49-s − 2·61-s + (1 + i)65-s + (−1 − i)73-s + (−1 − i)89-s + (−1 + i)97-s − 2i·101-s + (−1 + i)109-s + 2·113-s + ⋯ |
L(s) = 1 | + (1 − i)5-s + i·13-s − i·25-s + 2·29-s + (1 + i)37-s + (1 − i)41-s − i·49-s − 2·61-s + (1 + i)65-s + (−1 − i)73-s + (−1 − i)89-s + (−1 + i)97-s − 2i·101-s + (−1 + i)109-s + 2·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.600915766\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.600915766\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - iT \) |
good | 5 | \( 1 + (-1 + i)T - iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 2T + T^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 37 | \( 1 + (-1 - i)T + iT^{2} \) |
| 41 | \( 1 + (-1 + i)T - iT^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + 2T + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - iT^{2} \) |
| 73 | \( 1 + (1 + i)T + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (1 + i)T + iT^{2} \) |
| 97 | \( 1 + (1 - i)T - iT^{2} \) |
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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
−8.763181695814740425332622879866, −8.077196883974705767795534515826, −7.06536350557419186549458761571, −6.29981781133940215906406851320, −5.70167595478176793866327755513, −4.74554718864563633975788540722, −4.34554960100854779785093159264, −2.99740109841387028626714697373, −2.00928541277879096433162305045, −1.12451573249837154139486027186,
1.25156413959512857645501907717, 2.64578695447521666299827688073, 2.87017453069992817163761397677, 4.15017008628009042565288530091, 5.08644100637190729889763225675, 6.05006659469721442765779364500, 6.28538977692793145722632479705, 7.28467137467087155827139534592, 7.914363884419773255436589195893, 8.782447344429322762409291767914