L(s) = 1 | + (0.122 + 0.169i)2-s + (0.770 + 0.770i)3-s + (0.295 − 0.909i)4-s + (−0.0356 + 0.224i)6-s + (0.388 − 0.126i)8-s + 0.186i·9-s + (0.928 − 0.472i)12-s + (0.103 + 0.0163i)13-s + (−0.704 − 0.511i)16-s + (−0.0315 + 0.0229i)18-s + (−0.809 + 0.587i)23-s + (0.396 + 0.202i)24-s + (0.809 + 0.587i)25-s + (0.00993 + 0.0194i)26-s + (0.626 − 0.626i)27-s + ⋯ |
L(s) = 1 | + (0.122 + 0.169i)2-s + (0.770 + 0.770i)3-s + (0.295 − 0.909i)4-s + (−0.0356 + 0.224i)6-s + (0.388 − 0.126i)8-s + 0.186i·9-s + (0.928 − 0.472i)12-s + (0.103 + 0.0163i)13-s + (−0.704 − 0.511i)16-s + (−0.0315 + 0.0229i)18-s + (−0.809 + 0.587i)23-s + (0.396 + 0.202i)24-s + (0.809 + 0.587i)25-s + (0.00993 + 0.0194i)26-s + (0.626 − 0.626i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 943 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 943 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.441807599\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.441807599\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 + (0.809 - 0.587i)T \) |
| 41 | \( 1 + (0.104 + 0.994i)T \) |
good | 2 | \( 1 + (-0.122 - 0.169i)T + (-0.309 + 0.951i)T^{2} \) |
| 3 | \( 1 + (-0.770 - 0.770i)T + iT^{2} \) |
| 5 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 11 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 13 | \( 1 + (-0.103 - 0.0163i)T + (0.951 + 0.309i)T^{2} \) |
| 17 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 19 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 29 | \( 1 + (1.49 - 0.761i)T + (0.587 - 0.809i)T^{2} \) |
| 31 | \( 1 + (-0.128 - 0.395i)T + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 47 | \( 1 + (0.112 - 0.707i)T + (-0.951 - 0.309i)T^{2} \) |
| 53 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 71 | \( 1 + (0.705 - 1.38i)T + (-0.587 - 0.809i)T^{2} \) |
| 73 | \( 1 + 1.48iT - T^{2} \) |
| 79 | \( 1 - iT^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 97 | \( 1 + (0.587 - 0.809i)T^{2} \) |
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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
−10.21080877286324565183579193813, −9.380647167009585430071983908632, −8.934603913982091623358112769062, −7.76255583233376436597265894084, −6.85672364416859568704786712265, −5.87461978749524781712155621327, −5.01659908301187764365989226746, −4.00664192571358988147951389327, −3.01914642861865995209057695045, −1.65512285823835893651206760768,
1.86765192223819035797346627825, 2.69811017509965093664790945403, 3.68662332147519606169322804005, 4.74702191694468367763307408364, 6.19397249008780479112416962553, 7.03827276911629233861214934323, 7.86000461825369188206182958914, 8.284071464069863221998786216627, 9.159506489437112664199459871876, 10.26938621849042786673257482348