L(s) = 1 | + (−0.707 + 0.707i)3-s + (0.707 + 0.707i)7-s + 1.73i·11-s + (−1.22 − 1.22i)13-s + (−1.22 + 1.22i)17-s − 1.00·21-s + (−0.707 − 0.707i)27-s − i·29-s + (−1.22 − 1.22i)33-s + 1.73·39-s + (0.707 + 0.707i)47-s + 1.00i·49-s − 1.73i·51-s + (−1.22 + 1.22i)77-s − 1.73·79-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)3-s + (0.707 + 0.707i)7-s + 1.73i·11-s + (−1.22 − 1.22i)13-s + (−1.22 + 1.22i)17-s − 1.00·21-s + (−0.707 − 0.707i)27-s − i·29-s + (−1.22 − 1.22i)33-s + 1.73·39-s + (0.707 + 0.707i)47-s + 1.00i·49-s − 1.73i·51-s + (−1.22 + 1.22i)77-s − 1.73·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6369907140\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6369907140\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 11 | \( 1 - 1.73iT - T^{2} \) |
| 13 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 17 | \( 1 + (1.22 - 1.22i)T - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + 1.73T + T^{2} \) |
| 83 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
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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
−9.576165753675414741274589001943, −8.535805225931330388001800233568, −7.82020669366663350197912344885, −7.13932007756390817956122698738, −6.02218382482968823300801400304, −5.37863409658736358767383935530, −4.60149255541180347745763325253, −4.26017889430926682160828956344, −2.58689183562045310493675446873, −1.92136981210079176820927829416,
0.42642545648434157379426967415, 1.61796411544938937047138619822, 2.79390956894612696236904771830, 3.98009333154217250719782566298, 4.84373084308826641574571394368, 5.56606654833010718393678955630, 6.57082348345503599263214091606, 7.01180386577564204390458629700, 7.64539853857945811576696743457, 8.761417359660345184576516669838