| L(s) = 1 | − 0.879·5-s + 4.41·7-s − 1.04·11-s + 0.0418·13-s + 4.75·17-s − 19-s − 6.87·23-s − 4.22·25-s + 3.87·29-s − 7.78·31-s − 3.87·35-s + 1.22·37-s − 12.1·41-s − 2.55·43-s − 10.4·47-s + 12.4·49-s + 2.83·53-s + 0.916·55-s − 6.16·59-s − 5.95·61-s − 0.0368·65-s + 2.59·67-s − 6.71·71-s + 8.19·73-s − 4.59·77-s + 2.40·79-s − 12.1·83-s + ⋯ |
| L(s) = 1 | − 0.393·5-s + 1.66·7-s − 0.314·11-s + 0.0116·13-s + 1.15·17-s − 0.229·19-s − 1.43·23-s − 0.845·25-s + 0.720·29-s − 1.39·31-s − 0.655·35-s + 0.201·37-s − 1.89·41-s − 0.389·43-s − 1.52·47-s + 1.78·49-s + 0.389·53-s + 0.123·55-s − 0.802·59-s − 0.762·61-s − 0.00456·65-s + 0.317·67-s − 0.797·71-s + 0.958·73-s − 0.523·77-s + 0.270·79-s − 1.33·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 0.879T + 5T^{2} \) |
| 7 | \( 1 - 4.41T + 7T^{2} \) |
| 11 | \( 1 + 1.04T + 11T^{2} \) |
| 13 | \( 1 - 0.0418T + 13T^{2} \) |
| 17 | \( 1 - 4.75T + 17T^{2} \) |
| 23 | \( 1 + 6.87T + 23T^{2} \) |
| 29 | \( 1 - 3.87T + 29T^{2} \) |
| 31 | \( 1 + 7.78T + 31T^{2} \) |
| 37 | \( 1 - 1.22T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 + 2.55T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 2.83T + 53T^{2} \) |
| 59 | \( 1 + 6.16T + 59T^{2} \) |
| 61 | \( 1 + 5.95T + 61T^{2} \) |
| 67 | \( 1 - 2.59T + 67T^{2} \) |
| 71 | \( 1 + 6.71T + 71T^{2} \) |
| 73 | \( 1 - 8.19T + 73T^{2} \) |
| 79 | \( 1 - 2.40T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 8.80T + 89T^{2} \) |
| 97 | \( 1 + 1.14T + 97T^{2} \) |
| 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
−7.65590844046455785060479626059, −6.94665207063560643961350102771, −5.91530817585651186360466251033, −5.34301247799803253413457950437, −4.67887927870787700515864151990, −3.96807899752016426895071000300, −3.17970216638695217012050163057, −1.97934271108785529578081578217, −1.46171538438406033408741103536, 0,
1.46171538438406033408741103536, 1.97934271108785529578081578217, 3.17970216638695217012050163057, 3.96807899752016426895071000300, 4.67887927870787700515864151990, 5.34301247799803253413457950437, 5.91530817585651186360466251033, 6.94665207063560643961350102771, 7.65590844046455785060479626059
