L(s) = 1 | − 1.28·5-s + 2.78·7-s − 1.78·13-s + 4.86·17-s + 5.35·19-s − 7.57·23-s − 3.35·25-s − 5.28·29-s − 0.223·31-s − 3.57·35-s − 3.56·37-s − 6.29·41-s + 2.56·43-s − 4·47-s + 0.776·49-s + 2.71·53-s + 0.422·59-s + 1.77·61-s + 2.29·65-s − 6.78·67-s + 8.70·71-s − 12.9·73-s + 6.93·79-s − 9.13·83-s − 6.23·85-s − 16.8·89-s − 4.98·91-s + ⋯ |
L(s) = 1 | − 0.573·5-s + 1.05·7-s − 0.496·13-s + 1.17·17-s + 1.22·19-s − 1.58·23-s − 0.670·25-s − 0.980·29-s − 0.0400·31-s − 0.604·35-s − 0.586·37-s − 0.983·41-s + 0.391·43-s − 0.583·47-s + 0.110·49-s + 0.373·53-s + 0.0550·59-s + 0.227·61-s + 0.284·65-s − 0.829·67-s + 1.03·71-s − 1.51·73-s + 0.779·79-s − 1.00·83-s − 0.676·85-s − 1.78·89-s − 0.522·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + 1.28T + 5T^{2} \) |
| 7 | \( 1 - 2.78T + 7T^{2} \) |
| 13 | \( 1 + 1.78T + 13T^{2} \) |
| 17 | \( 1 - 4.86T + 17T^{2} \) |
| 19 | \( 1 - 5.35T + 19T^{2} \) |
| 23 | \( 1 + 7.57T + 23T^{2} \) |
| 29 | \( 1 + 5.28T + 29T^{2} \) |
| 31 | \( 1 + 0.223T + 31T^{2} \) |
| 37 | \( 1 + 3.56T + 37T^{2} \) |
| 41 | \( 1 + 6.29T + 41T^{2} \) |
| 43 | \( 1 - 2.56T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 2.71T + 53T^{2} \) |
| 59 | \( 1 - 0.422T + 59T^{2} \) |
| 61 | \( 1 - 1.77T + 61T^{2} \) |
| 67 | \( 1 + 6.78T + 67T^{2} \) |
| 71 | \( 1 - 8.70T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 - 6.93T + 79T^{2} \) |
| 83 | \( 1 + 9.13T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−7.63922650658534476383020161545, −6.95527665385914758434853130441, −5.80624154378289325870046393044, −5.40839578280128964132017252351, −4.62163319154328025950093548263, −3.84932490246872102358633709082, −3.21354628416874579819105785742, −2.06827145296304677783543263974, −1.32459012191785315455086949493, 0,
1.32459012191785315455086949493, 2.06827145296304677783543263974, 3.21354628416874579819105785742, 3.84932490246872102358633709082, 4.62163319154328025950093548263, 5.40839578280128964132017252351, 5.80624154378289325870046393044, 6.95527665385914758434853130441, 7.63922650658534476383020161545