L(s) = 1 | − 5-s − 4.24·11-s + 5.24·13-s − 4.24·17-s + 7·19-s − 4.24·23-s + 25-s + 10.2·29-s − 7.48·31-s − 5.24·37-s + 4.24·41-s − 5.24·43-s − 6·47-s + 8.48·53-s + 4.24·55-s − 1.75·59-s + 12.4·61-s − 5.24·65-s + 3.24·67-s − 12.7·71-s − 0.757·73-s + 11·79-s − 1.75·83-s + 4.24·85-s − 1.75·89-s − 7·95-s − 16.4·97-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.27·11-s + 1.45·13-s − 1.02·17-s + 1.60·19-s − 0.884·23-s + 0.200·25-s + 1.90·29-s − 1.34·31-s − 0.861·37-s + 0.662·41-s − 0.799·43-s − 0.875·47-s + 1.16·53-s + 0.572·55-s − 0.228·59-s + 1.59·61-s − 0.650·65-s + 0.396·67-s − 1.51·71-s − 0.0886·73-s + 1.23·79-s − 0.192·83-s + 0.460·85-s − 0.186·89-s − 0.718·95-s − 1.67·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.556375707\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.556375707\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 - 5.24T + 13T^{2} \) |
| 17 | \( 1 + 4.24T + 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 + 4.24T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 + 7.48T + 31T^{2} \) |
| 37 | \( 1 + 5.24T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 + 5.24T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 8.48T + 53T^{2} \) |
| 59 | \( 1 + 1.75T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 3.24T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 0.757T + 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 + 1.75T + 83T^{2} \) |
| 89 | \( 1 + 1.75T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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
−7.80299411096270061986814118464, −7.11028074128686601255734189813, −6.43368215861921039141253124270, −5.60058961235877587276937059915, −5.06577986215996851387848864635, −4.18070680872129161695802889610, −3.45512994672663051644891800793, −2.75191700138546762487572789827, −1.72247674039738335076559472967, −0.60611729130721679839025420884,
0.60611729130721679839025420884, 1.72247674039738335076559472967, 2.75191700138546762487572789827, 3.45512994672663051644891800793, 4.18070680872129161695802889610, 5.06577986215996851387848864635, 5.60058961235877587276937059915, 6.43368215861921039141253124270, 7.11028074128686601255734189813, 7.80299411096270061986814118464