L(s) = 1 | − 7-s + 2·13-s − 2·19-s + 6·29-s + 8·31-s − 4·37-s − 6·41-s + 2·43-s − 6·47-s + 49-s − 6·53-s + 12·59-s − 8·61-s + 2·67-s − 6·71-s − 2·73-s − 16·79-s − 6·89-s − 2·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 0.377·7-s + 0.554·13-s − 0.458·19-s + 1.11·29-s + 1.43·31-s − 0.657·37-s − 0.937·41-s + 0.304·43-s − 0.875·47-s + 1/7·49-s − 0.824·53-s + 1.56·59-s − 1.02·61-s + 0.244·67-s − 0.712·71-s − 0.234·73-s − 1.80·79-s − 0.635·89-s − 0.209·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.00725579998869, −13.51140623494615, −12.95876184545964, −12.66411703613028, −11.87603735376512, −11.72121685942583, −11.04336384291921, −10.46788099879211, −10.05123650091518, −9.714818520700761, −8.872230556591427, −8.532009991768569, −8.179772166475960, −7.396983070128892, −6.910585028489346, −6.309775989675187, −6.058914696149551, −5.273492371317213, −4.654316572929602, −4.259376137497891, −3.418050196978155, −3.058804669380094, −2.333044022710785, −1.580846174834889, −0.8856363250138436, 0,
0.8856363250138436, 1.580846174834889, 2.333044022710785, 3.058804669380094, 3.418050196978155, 4.259376137497891, 4.654316572929602, 5.273492371317213, 6.058914696149551, 6.309775989675187, 6.910585028489346, 7.396983070128892, 8.179772166475960, 8.532009991768569, 8.872230556591427, 9.714818520700761, 10.05123650091518, 10.46788099879211, 11.04336384291921, 11.72121685942583, 11.87603735376512, 12.66411703613028, 12.95876184545964, 13.51140623494615, 14.00725579998869