| L(s) = 1 | + 5-s + 7-s − 13-s − 6·17-s + 7·19-s + 4·23-s − 4·25-s + 3·29-s − 2·31-s + 35-s − 7·37-s + 8·41-s − 13·47-s + 49-s + 12·53-s − 59-s − 14·61-s − 65-s − 15·67-s + 8·71-s + 73-s + 4·79-s − 14·83-s − 6·85-s + 6·89-s − 91-s + 7·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 0.277·13-s − 1.45·17-s + 1.60·19-s + 0.834·23-s − 4/5·25-s + 0.557·29-s − 0.359·31-s + 0.169·35-s − 1.15·37-s + 1.24·41-s − 1.89·47-s + 1/7·49-s + 1.64·53-s − 0.130·59-s − 1.79·61-s − 0.124·65-s − 1.83·67-s + 0.949·71-s + 0.117·73-s + 0.450·79-s − 1.53·83-s − 0.650·85-s + 0.635·89-s − 0.104·91-s + 0.718·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.298115739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.298115739\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.08072928150453, −13.84912483357683, −13.38154201120983, −12.86173439442426, −12.23707279677831, −11.69587038807738, −11.31533682882981, −10.73583399363254, −10.23550288557732, −9.629245215759002, −9.114272356542122, −8.816843719742649, −7.969354033436067, −7.587970484765770, −6.877102385810350, −6.564694387538403, −5.620624397093421, −5.411414270065303, −4.609120130967076, −4.235076902638327, −3.261622028621039, −2.826481895656701, −1.974660461404856, −1.473127565785344, −0.5106573227329011,
0.5106573227329011, 1.473127565785344, 1.974660461404856, 2.826481895656701, 3.261622028621039, 4.235076902638327, 4.609120130967076, 5.411414270065303, 5.620624397093421, 6.564694387538403, 6.877102385810350, 7.587970484765770, 7.969354033436067, 8.816843719742649, 9.114272356542122, 9.629245215759002, 10.23550288557732, 10.73583399363254, 11.31533682882981, 11.69587038807738, 12.23707279677831, 12.86173439442426, 13.38154201120983, 13.84912483357683, 14.08072928150453
