L(s) = 1 | + 1.23·5-s + 2.35·7-s − 6.15·11-s + 13-s − 6.47·17-s + 5.25·19-s − 3.47·25-s − 4·29-s + 2.35·31-s + 2.90·35-s − 10.9·37-s − 5.23·41-s − 7.60·43-s + 9.06·47-s − 1.47·49-s + 4.94·53-s − 7.60·55-s − 9.06·59-s + 4.47·61-s + 1.23·65-s − 14.6·67-s − 6.15·71-s + 6·73-s − 14.4·77-s + 15.2·79-s + 3.24·83-s − 8.00·85-s + ⋯ |
L(s) = 1 | + 0.552·5-s + 0.888·7-s − 1.85·11-s + 0.277·13-s − 1.56·17-s + 1.20·19-s − 0.694·25-s − 0.742·29-s + 0.422·31-s + 0.491·35-s − 1.79·37-s − 0.817·41-s − 1.16·43-s + 1.32·47-s − 0.210·49-s + 0.679·53-s − 1.02·55-s − 1.17·59-s + 0.572·61-s + 0.153·65-s − 1.79·67-s − 0.730·71-s + 0.702·73-s − 1.64·77-s + 1.71·79-s + 0.356·83-s − 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 - 2.35T + 7T^{2} \) |
| 11 | \( 1 + 6.15T + 11T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 - 5.25T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 2.35T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 5.23T + 41T^{2} \) |
| 43 | \( 1 + 7.60T + 43T^{2} \) |
| 47 | \( 1 - 9.06T + 47T^{2} \) |
| 53 | \( 1 - 4.94T + 53T^{2} \) |
| 59 | \( 1 + 9.06T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 + 6.15T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 - 3.24T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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
−8.137031078416664326979206132156, −7.49775279124804829481831584310, −6.74064143749351907963785102476, −5.65840242197305756977922298622, −5.22533221144118838232252844961, −4.51248984334055485640508716315, −3.33822833955168680429139700273, −2.35805585901229559412536340714, −1.64817425650453540421541563952, 0,
1.64817425650453540421541563952, 2.35805585901229559412536340714, 3.33822833955168680429139700273, 4.51248984334055485640508716315, 5.22533221144118838232252844961, 5.65840242197305756977922298622, 6.74064143749351907963785102476, 7.49775279124804829481831584310, 8.137031078416664326979206132156