L(s) = 1 | + i·3-s + (1.96 − 1.06i)5-s + 3.03i·7-s − 9-s + 2.01·11-s − 1.13i·13-s + (1.06 + 1.96i)15-s − 0.286i·17-s − 1.78·19-s − 3.03·21-s − 7.18i·23-s + (2.71 − 4.19i)25-s − i·27-s + 8.82·29-s + 2.29·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (0.878 − 0.477i)5-s + 1.14i·7-s − 0.333·9-s + 0.607·11-s − 0.316i·13-s + (0.275 + 0.507i)15-s − 0.0693i·17-s − 0.408·19-s − 0.661·21-s − 1.49i·23-s + (0.543 − 0.839i)25-s − 0.192i·27-s + 1.63·29-s + 0.412·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.433374591\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.433374591\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-1.96 + 1.06i)T \) |
| 67 | \( 1 - iT \) |
good | 7 | \( 1 - 3.03iT - 7T^{2} \) |
| 11 | \( 1 - 2.01T + 11T^{2} \) |
| 13 | \( 1 + 1.13iT - 13T^{2} \) |
| 17 | \( 1 + 0.286iT - 17T^{2} \) |
| 19 | \( 1 + 1.78T + 19T^{2} \) |
| 23 | \( 1 + 7.18iT - 23T^{2} \) |
| 29 | \( 1 - 8.82T + 29T^{2} \) |
| 31 | \( 1 - 2.29T + 31T^{2} \) |
| 37 | \( 1 + 2.36iT - 37T^{2} \) |
| 41 | \( 1 - 5.15T + 41T^{2} \) |
| 43 | \( 1 - 2.11iT - 43T^{2} \) |
| 47 | \( 1 - 6.00iT - 47T^{2} \) |
| 53 | \( 1 + 9.22iT - 53T^{2} \) |
| 59 | \( 1 - 5.81T + 59T^{2} \) |
| 61 | \( 1 - 7.84T + 61T^{2} \) |
| 71 | \( 1 - 5.24T + 71T^{2} \) |
| 73 | \( 1 - 8.12iT - 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 - 9.42iT - 83T^{2} \) |
| 89 | \( 1 - 1.93T + 89T^{2} \) |
| 97 | \( 1 + 5.99iT - 97T^{2} \) |
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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
−8.619424887308529275926685344234, −8.168082011326312123425483932918, −6.71828428885516372396512772101, −6.23154767815202346668974489965, −5.49376701073443031711272567402, −4.81752954096434130266807807799, −4.08829474368878240609888116350, −2.77740556660058258882459851399, −2.28816209694914649058411020391, −0.919922210727132227727114419108,
0.947537795720161870711994608971, 1.75701349095791856001780365091, 2.80128934230337973655240993982, 3.72340316240704040115128312261, 4.57347872191224689726967898128, 5.60383787719934154707189578710, 6.33746930304025812219124847784, 6.93084894898973541347335791485, 7.42113697170902796915811058131, 8.348443891474246056125442050329