| L(s) = 1 | − 5-s + 5·11-s − 2·13-s + 6·17-s + 2·19-s − 6·23-s − 4·25-s + 3·29-s + 5·31-s − 2·37-s + 8·41-s + 4·43-s − 4·47-s − 9·53-s − 5·55-s + 3·59-s + 12·61-s + 2·65-s − 2·67-s − 8·71-s + 14·73-s − 79-s − 17·83-s − 6·85-s + 18·89-s − 2·95-s − 3·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.50·11-s − 0.554·13-s + 1.45·17-s + 0.458·19-s − 1.25·23-s − 4/5·25-s + 0.557·29-s + 0.898·31-s − 0.328·37-s + 1.24·41-s + 0.609·43-s − 0.583·47-s − 1.23·53-s − 0.674·55-s + 0.390·59-s + 1.53·61-s + 0.248·65-s − 0.244·67-s − 0.949·71-s + 1.63·73-s − 0.112·79-s − 1.86·83-s − 0.650·85-s + 1.90·89-s − 0.205·95-s − 0.304·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.034649761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.034649761\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−7.87559219771508266000657031536, −7.37397913531658989036753357633, −6.48365821380494066075340362895, −5.94747060999875640798234393556, −5.09727835738127525774038667366, −4.17603327850694321278899797062, −3.71651364150977666215109777015, −2.81204094769716594006356009492, −1.69472917148202922072753031703, −0.75735217169580535127033287965,
0.75735217169580535127033287965, 1.69472917148202922072753031703, 2.81204094769716594006356009492, 3.71651364150977666215109777015, 4.17603327850694321278899797062, 5.09727835738127525774038667366, 5.94747060999875640798234393556, 6.48365821380494066075340362895, 7.37397913531658989036753357633, 7.87559219771508266000657031536
