L(s) = 1 | + (−0.382 + 0.923i)2-s + 0.765i·3-s + (−0.707 − 0.707i)4-s − i·5-s + (−0.707 − 0.292i)6-s + (0.923 − 0.382i)8-s + 0.414·9-s + (0.923 + 0.382i)10-s − 1.41i·11-s + (0.541 − 0.541i)12-s − 1.84i·13-s + 0.765·15-s + i·16-s + (−0.158 + 0.382i)18-s + i·19-s + (−0.707 + 0.707i)20-s + ⋯ |
L(s) = 1 | + (−0.382 + 0.923i)2-s + 0.765i·3-s + (−0.707 − 0.707i)4-s − i·5-s + (−0.707 − 0.292i)6-s + (0.923 − 0.382i)8-s + 0.414·9-s + (0.923 + 0.382i)10-s − 1.41i·11-s + (0.541 − 0.541i)12-s − 1.84i·13-s + 0.765·15-s + i·16-s + (−0.158 + 0.382i)18-s + i·19-s + (−0.707 + 0.707i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7652315986\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7652315986\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.382 - 0.923i)T \) |
| 5 | \( 1 + iT \) |
| 19 | \( 1 - iT \) |
good | 3 | \( 1 - 0.765iT - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 + 1.84iT - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 0.765iT - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 0.765iT - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - 1.41iT - T^{2} \) |
| 67 | \( 1 - 1.84iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 1.84T + T^{2} \) |
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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
−10.22467616710998489542562032798, −9.774525656887856241351642908736, −8.592753052261509320041223596891, −8.327613216072979561010171038131, −7.32190560437348964866597178948, −5.81730222421585674546816566393, −5.51542614978338252233282019947, −4.43419599092851498938453175301, −3.44280843102116869893743109298, −1.04323228490519563664463447560,
1.78344252486711102387075959893, 2.42085619131412664265747652184, 3.92515368015775189451262716555, 4.71721524418332331860155676176, 6.55009617606950544689008705295, 7.12342606537913083977318859501, 7.73446317341803541229733829064, 9.118886733049526694471597905605, 9.645858487647965762154122222754, 10.55044517793837874624863700973