L(s) = 1 | + 3-s + 2·5-s + 4·7-s + 9-s − 11-s + 6·13-s + 2·15-s + 8·19-s + 4·21-s − 25-s + 27-s + 6·29-s − 33-s + 8·35-s − 6·37-s + 6·39-s + 10·41-s + 8·43-s + 2·45-s + 9·49-s + 6·53-s − 2·55-s + 8·57-s − 4·59-s + 2·61-s + 4·63-s + 12·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s − 0.301·11-s + 1.66·13-s + 0.516·15-s + 1.83·19-s + 0.872·21-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.174·33-s + 1.35·35-s − 0.986·37-s + 0.960·39-s + 1.56·41-s + 1.21·43-s + 0.298·45-s + 9/7·49-s + 0.824·53-s − 0.269·55-s + 1.05·57-s − 0.520·59-s + 0.256·61-s + 0.503·63-s + 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.091574870\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.091574870\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.43048815002828, −13.09918350820770, −12.33055522177635, −11.88006057990819, −11.35453146526882, −10.91313781115525, −10.50653909999357, −9.962443243302874, −9.397721384395422, −8.950105060695698, −8.513559651426019, −8.006196617817753, −7.618289004168358, −7.096977473260241, −6.362707085060903, −5.783994878223610, −5.444288712315609, −4.904132554716384, −4.248342858819457, −3.732712555367558, −3.052732225904173, −2.443056890765215, −1.865496810306989, −1.178758691426578, −0.9529533387144505,
0.9529533387144505, 1.178758691426578, 1.865496810306989, 2.443056890765215, 3.052732225904173, 3.732712555367558, 4.248342858819457, 4.904132554716384, 5.444288712315609, 5.783994878223610, 6.362707085060903, 7.096977473260241, 7.618289004168358, 8.006196617817753, 8.513559651426019, 8.950105060695698, 9.397721384395422, 9.962443243302874, 10.50653909999357, 10.91313781115525, 11.35453146526882, 11.88006057990819, 12.33055522177635, 13.09918350820770, 13.43048815002828