| L(s) = 1 | − 3-s − 2·7-s − 2·9-s + 13-s + 6·17-s + 2·19-s + 2·21-s − 5·25-s + 5·27-s + 3·29-s + 5·31-s + 8·37-s − 39-s + 3·41-s + 8·43-s + 9·47-s − 3·49-s − 6·51-s + 6·53-s − 2·57-s + 12·59-s + 14·61-s + 4·63-s + 8·67-s − 15·71-s − 7·73-s + 5·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s − 2/3·9-s + 0.277·13-s + 1.45·17-s + 0.458·19-s + 0.436·21-s − 25-s + 0.962·27-s + 0.557·29-s + 0.898·31-s + 1.31·37-s − 0.160·39-s + 0.468·41-s + 1.21·43-s + 1.31·47-s − 3/7·49-s − 0.840·51-s + 0.824·53-s − 0.264·57-s + 1.56·59-s + 1.79·61-s + 0.503·63-s + 0.977·67-s − 1.78·71-s − 0.819·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.780389422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.780389422\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−14.83356918431257, −14.50942368396609, −13.95015739221165, −13.35873639089483, −12.91409514911685, −12.10342950940953, −11.94783832541857, −11.39698673939049, −10.71786208031768, −10.17101675544418, −9.728109036120614, −9.185002750063967, −8.467244419334409, −7.907014017558513, −7.393275693789096, −6.581156902318762, −6.131997926156398, −5.564764581265177, −5.229071402498223, −4.165860200155503, −3.732151521129691, −2.839478952491826, −2.471521227182789, −1.106635877864263, −0.6193102953133714,
0.6193102953133714, 1.106635877864263, 2.471521227182789, 2.839478952491826, 3.732151521129691, 4.165860200155503, 5.229071402498223, 5.564764581265177, 6.131997926156398, 6.581156902318762, 7.393275693789096, 7.907014017558513, 8.467244419334409, 9.185002750063967, 9.728109036120614, 10.17101675544418, 10.71786208031768, 11.39698673939049, 11.94783832541857, 12.10342950940953, 12.91409514911685, 13.35873639089483, 13.95015739221165, 14.50942368396609, 14.83356918431257
