| L(s) = 1 | − 3-s − 5-s + 4·7-s + 9-s + 4·11-s + 2·13-s + 15-s − 2·17-s + 19-s − 4·21-s − 8·23-s + 25-s − 27-s − 6·29-s + 4·31-s − 4·33-s − 4·35-s + 10·37-s − 2·39-s − 2·41-s − 12·43-s − 45-s + 9·49-s + 2·51-s − 6·53-s − 4·55-s − 57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.258·15-s − 0.485·17-s + 0.229·19-s − 0.872·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.696·33-s − 0.676·35-s + 1.64·37-s − 0.320·39-s − 0.312·41-s − 1.82·43-s − 0.149·45-s + 9/7·49-s + 0.280·51-s − 0.824·53-s − 0.539·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.122025533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.122025533\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−15.88178310734640, −15.19266493238460, −14.55875943180263, −14.41249853893681, −13.54816861546307, −13.11548102379710, −12.20490676056008, −11.72813450369547, −11.40197952366597, −11.10864374776156, −10.20855786264491, −9.719770141945324, −8.868932241848153, −8.372352369710144, −7.825619281856169, −7.266994055829211, −6.386494502175008, −6.049164308492540, −5.169157898713591, −4.540537094246793, −4.076979750077964, −3.405610713848296, −2.071153038459897, −1.573650720362241, −0.6654711466612581,
0.6654711466612581, 1.573650720362241, 2.071153038459897, 3.405610713848296, 4.076979750077964, 4.540537094246793, 5.169157898713591, 6.049164308492540, 6.386494502175008, 7.266994055829211, 7.825619281856169, 8.372352369710144, 8.868932241848153, 9.719770141945324, 10.20855786264491, 11.10864374776156, 11.40197952366597, 11.72813450369547, 12.20490676056008, 13.11548102379710, 13.54816861546307, 14.41249853893681, 14.55875943180263, 15.19266493238460, 15.88178310734640
