| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s + 4·11-s + 15-s − 2·17-s + 6·19-s + 2·21-s + 25-s − 27-s + 10·29-s + 2·31-s − 4·33-s + 2·35-s + 8·37-s + 2·41-s + 4·43-s − 45-s − 8·47-s − 3·49-s + 2·51-s + 2·53-s − 4·55-s − 6·57-s + 10·61-s − 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.258·15-s − 0.485·17-s + 1.37·19-s + 0.436·21-s + 1/5·25-s − 0.192·27-s + 1.85·29-s + 0.359·31-s − 0.696·33-s + 0.338·35-s + 1.31·37-s + 0.312·41-s + 0.609·43-s − 0.149·45-s − 1.16·47-s − 3/7·49-s + 0.280·51-s + 0.274·53-s − 0.539·55-s − 0.794·57-s + 1.28·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.760601726\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.760601726\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 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
−15.91597633821424, −15.11249973194527, −14.62418604010196, −13.95077422004835, −13.48769665739612, −12.80557765268995, −12.25280173810947, −11.77345809867181, −11.41089984472388, −10.74062187006921, −10.03071161493983, −9.515467816853785, −9.120944793815429, −8.237297007601071, −7.741849356497579, −6.795688376253523, −6.626386468384000, −6.004365162256117, −5.155391448699053, −4.536066016931771, −3.880972091576372, −3.228552839524676, −2.467620342140016, −1.232035567649132, −0.6545249083595234,
0.6545249083595234, 1.232035567649132, 2.467620342140016, 3.228552839524676, 3.880972091576372, 4.536066016931771, 5.155391448699053, 6.004365162256117, 6.626386468384000, 6.795688376253523, 7.741849356497579, 8.237297007601071, 9.120944793815429, 9.515467816853785, 10.03071161493983, 10.74062187006921, 11.41089984472388, 11.77345809867181, 12.25280173810947, 12.80557765268995, 13.48769665739612, 13.95077422004835, 14.62418604010196, 15.11249973194527, 15.91597633821424
