| L(s) = 1 | − 2-s + 4-s − 8-s − 3·11-s + 5·13-s + 16-s − 5·19-s + 3·22-s − 9·23-s − 5·26-s + 10·31-s − 32-s + 37-s + 5·38-s + 9·41-s − 8·43-s − 3·44-s + 9·46-s − 3·47-s + 5·52-s − 3·53-s + 12·59-s − 8·61-s − 10·62-s + 64-s − 8·67-s + 6·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.904·11-s + 1.38·13-s + 1/4·16-s − 1.14·19-s + 0.639·22-s − 1.87·23-s − 0.980·26-s + 1.79·31-s − 0.176·32-s + 0.164·37-s + 0.811·38-s + 1.40·41-s − 1.21·43-s − 0.452·44-s + 1.32·46-s − 0.437·47-s + 0.693·52-s − 0.412·53-s + 1.56·59-s − 1.02·61-s − 1.27·62-s + 1/8·64-s − 0.977·67-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−15.87263719872015, −15.44807808082812, −14.89852602029061, −14.10683861572551, −13.66701439980517, −13.05308288628232, −12.57644130259942, −11.80031847920001, −11.41761794197327, −10.68920106863785, −10.29789596209440, −9.876786375747484, −9.030407244315660, −8.499706883323908, −7.995743666390454, −7.698075857535586, −6.550623840469268, −6.305720710209231, −5.722877985481717, −4.796963439828814, −4.106199053016337, −3.409804388684531, −2.521142314380311, −1.936557030495509, −0.9775315649181988, 0,
0.9775315649181988, 1.936557030495509, 2.521142314380311, 3.409804388684531, 4.106199053016337, 4.796963439828814, 5.722877985481717, 6.305720710209231, 6.550623840469268, 7.698075857535586, 7.995743666390454, 8.499706883323908, 9.030407244315660, 9.876786375747484, 10.29789596209440, 10.68920106863785, 11.41761794197327, 11.80031847920001, 12.57644130259942, 13.05308288628232, 13.66701439980517, 14.10683861572551, 14.89852602029061, 15.44807808082812, 15.87263719872015
