L(s) = 1 | + (0.923 + 0.382i)2-s + (0.707 + 0.707i)4-s + (0.923 − 0.382i)5-s + (0.382 + 0.923i)8-s + 10-s + i·16-s − 0.765·17-s + (−0.707 − 0.292i)19-s + (0.923 + 0.382i)20-s + (−1.30 − 1.30i)23-s + (0.707 − 0.707i)25-s + 1.41i·31-s + (−0.382 + 0.923i)32-s + (−0.707 − 0.292i)34-s + (−0.541 − 0.541i)38-s + ⋯ |
L(s) = 1 | + (0.923 + 0.382i)2-s + (0.707 + 0.707i)4-s + (0.923 − 0.382i)5-s + (0.382 + 0.923i)8-s + 10-s + i·16-s − 0.765·17-s + (−0.707 − 0.292i)19-s + (0.923 + 0.382i)20-s + (−1.30 − 1.30i)23-s + (0.707 − 0.707i)25-s + 1.41i·31-s + (−0.382 + 0.923i)32-s + (−0.707 − 0.292i)34-s + (−0.541 − 0.541i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.105641431\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.105641431\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.923 - 0.382i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.923 + 0.382i)T \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 17 | \( 1 + 0.765T + T^{2} \) |
| 19 | \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + (1.30 + 1.30i)T + iT^{2} \) |
| 29 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 - 1.41iT - T^{2} \) |
| 37 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + 0.765iT - T^{2} \) |
| 53 | \( 1 + (0.541 + 1.30i)T + (-0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 61 | \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + 2T + T^{2} \) |
| 83 | \( 1 + (0.541 - 1.30i)T + (-0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.905204395800722353004664386151, −8.670698693894857494767062924708, −8.363635399281638847200438039693, −7.00518955941507373689594596181, −6.45639434421676765947235905880, −5.67033363382831265693362083510, −4.79681157729173774032663385377, −4.08396415071599833622334062136, −2.73398938009807030052816169057, −1.88790653972637559442158164815,
1.72936354184101694682411560930, 2.48603981667718517981696148874, 3.65056406088643831899936314046, 4.51886477065800443228240645324, 5.64230207406569749044529889509, 6.10164898894360141079064368225, 6.94221917818555594715019020357, 7.88314529120052279626759985134, 9.135535005645679744037285055809, 9.870093125098831612107463924372