L(s) = 1 | + (1.41 + 1.41i)5-s + 7-s + (1.89 − 1.89i)11-s + (0.853 + 0.853i)13-s + 2.59i·17-s + 0.206i·23-s − 0.999i·25-s + (3.10 − 3.10i)29-s − 1.70i·31-s + (1.41 + 1.41i)35-s + (−1 + i)37-s + 11.0·41-s + (7.12 + 7.12i)43-s − 3.24·47-s + 49-s + ⋯ |
L(s) = 1 | + (0.632 + 0.632i)5-s + 0.377·7-s + (0.572 − 0.572i)11-s + (0.236 + 0.236i)13-s + 0.628i·17-s + 0.0431i·23-s − 0.199i·25-s + (0.576 − 0.576i)29-s − 0.306i·31-s + (0.239 + 0.239i)35-s + (−0.164 + 0.164i)37-s + 1.72·41-s + (1.08 + 1.08i)43-s − 0.472·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.488349143\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.488349143\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + (-1.41 - 1.41i)T + 5iT^{2} \) |
| 11 | \( 1 + (-1.89 + 1.89i)T - 11iT^{2} \) |
| 13 | \( 1 + (-0.853 - 0.853i)T + 13iT^{2} \) |
| 17 | \( 1 - 2.59iT - 17T^{2} \) |
| 19 | \( 1 - 19iT^{2} \) |
| 23 | \( 1 - 0.206iT - 23T^{2} \) |
| 29 | \( 1 + (-3.10 + 3.10i)T - 29iT^{2} \) |
| 31 | \( 1 + 1.70iT - 31T^{2} \) |
| 37 | \( 1 + (1 - i)T - 37iT^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + (-7.12 - 7.12i)T + 43iT^{2} \) |
| 47 | \( 1 + 3.24T + 47T^{2} \) |
| 53 | \( 1 + (-0.722 - 0.722i)T + 53iT^{2} \) |
| 59 | \( 1 + (-5.00 + 5.00i)T - 59iT^{2} \) |
| 61 | \( 1 + (3.14 + 3.14i)T + 61iT^{2} \) |
| 67 | \( 1 + (4.14 - 4.14i)T - 67iT^{2} \) |
| 71 | \( 1 - 10.0iT - 71T^{2} \) |
| 73 | \( 1 + 5.66iT - 73T^{2} \) |
| 79 | \( 1 - 3.41iT - 79T^{2} \) |
| 83 | \( 1 + (7.27 + 7.27i)T + 83iT^{2} \) |
| 89 | \( 1 - 17.1T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.463682649567083483775284850469, −7.82116215086521825208645198123, −6.92468358243631604364874131536, −6.12809432745204108284327654430, −5.86017600983902621581993800679, −4.63500624147138295523503024363, −3.93058214996612992589164343195, −2.92169242564145315767383705737, −2.09135353004027991184080017082, −1.00671678273470502183618657189,
0.905320382669864793567325801441, 1.79094298529089141311874978066, 2.77587813683725636525739466410, 3.91786644519296918786672203224, 4.70269458287180833492725589719, 5.38389695818395172318156790849, 6.07718452255871315085625614743, 7.00456088387680390415260936601, 7.60164872023816403320219904352, 8.550104013135351862729349596287