L(s) = 1 | + (1 − i)5-s − 9-s − i·13-s + 2i·17-s − i·25-s + (1 + i)37-s + (−1 + i)41-s + (−1 + i)45-s − i·49-s − 2·53-s − 2·61-s + (−1 − i)65-s + (1 + i)73-s + 81-s + (2 + 2i)85-s + ⋯ |
L(s) = 1 | + (1 − i)5-s − 9-s − i·13-s + 2i·17-s − i·25-s + (1 + i)37-s + (−1 + i)41-s + (−1 + i)45-s − i·49-s − 2·53-s − 2·61-s + (−1 − i)65-s + (1 + i)73-s + 81-s + (2 + 2i)85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9133529951\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9133529951\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 + iT \) |
good | 3 | \( 1 + T^{2} \) |
| 5 | \( 1 + (-1 + i)T - iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 17 | \( 1 - 2iT - T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 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 | \( 1 + 2T + T^{2} \) |
| 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
−11.31069611276589590934868735387, −10.35723704972721346096764210950, −9.566869835608252480345014322619, −8.511196947928718310135508737938, −8.078498275041504390228503343954, −6.27893573026064318113826885391, −5.72025115057794121051462155357, −4.71912387984632983114618621555, −3.16866821116746938307111895430, −1.63117168589590306032425342219,
2.23985557201118057159815943019, 3.15044123081260147708106111956, 4.84197831786097185235208915102, 5.93769959301795202666695387774, 6.71394351084427805407794498246, 7.66184863277729996560368845822, 9.133629002780230953314977305340, 9.522346323367501895718308396668, 10.72184922602421536327586719363, 11.35159825788314744739280946884