| L(s) = 1 | − 3·3-s − 5-s + 6·9-s + 5·11-s − 13-s + 3·15-s + 8·17-s − 4·19-s − 23-s + 25-s − 9·27-s − 6·29-s − 31-s − 15·33-s + 7·37-s + 3·39-s + 7·41-s + 6·43-s − 6·45-s + 3·47-s − 24·51-s + 6·53-s − 5·55-s + 12·57-s + 5·61-s + 65-s − 15·67-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.447·5-s + 2·9-s + 1.50·11-s − 0.277·13-s + 0.774·15-s + 1.94·17-s − 0.917·19-s − 0.208·23-s + 1/5·25-s − 1.73·27-s − 1.11·29-s − 0.179·31-s − 2.61·33-s + 1.15·37-s + 0.480·39-s + 1.09·41-s + 0.914·43-s − 0.894·45-s + 0.437·47-s − 3.36·51-s + 0.824·53-s − 0.674·55-s + 1.58·57-s + 0.640·61-s + 0.124·65-s − 1.83·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.300957810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.300957810\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−15.34560039139475, −14.80944443264359, −14.47684804199936, −13.73153738689180, −12.83003065502844, −12.49052188078403, −12.03005216402508, −11.65597082743532, −11.05870452236723, −10.68319293168274, −9.935758437326993, −9.508437285914628, −8.861661415138790, −7.905241359633777, −7.422242266506847, −6.884841829004096, −6.128722025652305, −5.844935035818086, −5.246959929712909, −4.309615343388906, −4.103481246157634, −3.278143656863629, −2.047668506315882, −1.122855135949429, −0.6126373728720444,
0.6126373728720444, 1.122855135949429, 2.047668506315882, 3.278143656863629, 4.103481246157634, 4.309615343388906, 5.246959929712909, 5.844935035818086, 6.128722025652305, 6.884841829004096, 7.422242266506847, 7.905241359633777, 8.861661415138790, 9.508437285914628, 9.935758437326993, 10.68319293168274, 11.05870452236723, 11.65597082743532, 12.03005216402508, 12.49052188078403, 12.83003065502844, 13.73153738689180, 14.47684804199936, 14.80944443264359, 15.34560039139475
