| L(s) = 1 | − 7·3-s + 7·7-s + 22·9-s − 7·11-s − 3·13-s − 61·17-s + 48·19-s − 49·21-s − 58·23-s + 35·27-s + 219·29-s + 298·31-s + 49·33-s + 170·37-s + 21·39-s + 50·41-s − 484·43-s − 131·47-s + 49·49-s + 427·51-s − 210·53-s − 336·57-s − 782·59-s + 488·61-s + 154·63-s − 494·67-s + 406·69-s + ⋯ |
| L(s) = 1 | − 1.34·3-s + 0.377·7-s + 0.814·9-s − 0.191·11-s − 0.0640·13-s − 0.870·17-s + 0.579·19-s − 0.509·21-s − 0.525·23-s + 0.249·27-s + 1.40·29-s + 1.72·31-s + 0.258·33-s + 0.755·37-s + 0.0862·39-s + 0.190·41-s − 1.71·43-s − 0.406·47-s + 1/7·49-s + 1.17·51-s − 0.544·53-s − 0.780·57-s − 1.72·59-s + 1.02·61-s + 0.307·63-s − 0.900·67-s + 0.708·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( 1 + 7 T + p^{3} T^{2} \) |
| 11 | \( 1 + 7 T + p^{3} T^{2} \) |
| 13 | \( 1 + 3 T + p^{3} T^{2} \) |
| 17 | \( 1 + 61 T + p^{3} T^{2} \) |
| 19 | \( 1 - 48 T + p^{3} T^{2} \) |
| 23 | \( 1 + 58 T + p^{3} T^{2} \) |
| 29 | \( 1 - 219 T + p^{3} T^{2} \) |
| 31 | \( 1 - 298 T + p^{3} T^{2} \) |
| 37 | \( 1 - 170 T + p^{3} T^{2} \) |
| 41 | \( 1 - 50 T + p^{3} T^{2} \) |
| 43 | \( 1 + 484 T + p^{3} T^{2} \) |
| 47 | \( 1 + 131 T + p^{3} T^{2} \) |
| 53 | \( 1 + 210 T + p^{3} T^{2} \) |
| 59 | \( 1 + 782 T + p^{3} T^{2} \) |
| 61 | \( 1 - 8 p T + p^{3} T^{2} \) |
| 67 | \( 1 + 494 T + p^{3} T^{2} \) |
| 71 | \( 1 + 240 T + p^{3} T^{2} \) |
| 73 | \( 1 + 58 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1065 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1036 T + p^{3} T^{2} \) |
| 89 | \( 1 - 608 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1339 T + p^{3} T^{2} \) |
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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
−9.961648579108541590530163162551, −8.706205754026126333519495680957, −7.83410753476126920214675159652, −6.67178988323883805070741709260, −6.10569258626740417124170733859, −5.01616750526410377050193877277, −4.44064820003552335328788621536, −2.80702106357602276264079818460, −1.25236998558974372685885174302, 0,
1.25236998558974372685885174302, 2.80702106357602276264079818460, 4.44064820003552335328788621536, 5.01616750526410377050193877277, 6.10569258626740417124170733859, 6.67178988323883805070741709260, 7.83410753476126920214675159652, 8.706205754026126333519495680957, 9.961648579108541590530163162551
