L(s) = 1 | − i·2-s − 4-s + i·8-s + 16-s + (1 + i)17-s + (−1 − i)19-s + (1 + i)23-s − i·32-s + (1 − i)34-s + (−1 + i)38-s + (1 − i)46-s + (1 + i)47-s + i·49-s + 2·53-s + (1 − i)61-s + ⋯ |
L(s) = 1 | − i·2-s − 4-s + i·8-s + 16-s + (1 + i)17-s + (−1 − i)19-s + (1 + i)23-s − i·32-s + (1 − i)34-s + (−1 + i)38-s + (1 − i)46-s + (1 + i)47-s + i·49-s + 2·53-s + (1 − i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.144775467\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.144775467\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (-1 - i)T + iT^{2} \) |
| 19 | \( 1 + (1 + i)T + iT^{2} \) |
| 23 | \( 1 + (-1 - i)T + iT^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-1 - i)T + iT^{2} \) |
| 53 | \( 1 - 2T + T^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + (-1 + i)T - iT^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + 2iT - T^{2} \) |
| 83 | \( 1 + 2iT - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−8.924840331713877732469922041288, −8.012384872896461317751539076957, −7.32857858289337515784577252081, −6.22559801106408865588512433282, −5.46057489774343533994656622524, −4.65177911743131635148143819386, −3.83710651073430088809657606669, −3.05542926507876727353684843455, −2.10842343207064511430585286330, −1.03148092965203080579482246530,
0.898819031395489928031866056968, 2.48480195480771585385219921642, 3.63399016935119989243603171591, 4.34747779460662295910268534792, 5.32900669016639210065358322191, 5.74601958566405836817432535151, 6.89490832464134106515289714728, 7.09624950196721262391253256446, 8.237971496252668051365799766698, 8.526233545765959870534761029644