| L(s) = 1 | + 2·3-s + 7-s + 9-s − 3·11-s + 13-s − 3·17-s − 2·19-s + 2·21-s + 3·23-s − 5·25-s − 4·27-s + 4·31-s − 6·33-s + 2·37-s + 2·39-s + 6·41-s − 8·43-s + 6·47-s − 6·49-s − 6·51-s − 4·57-s − 61-s + 63-s − 14·67-s + 6·69-s + 6·71-s + 2·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 0.904·11-s + 0.277·13-s − 0.727·17-s − 0.458·19-s + 0.436·21-s + 0.625·23-s − 25-s − 0.769·27-s + 0.718·31-s − 1.04·33-s + 0.328·37-s + 0.320·39-s + 0.937·41-s − 1.21·43-s + 0.875·47-s − 6/7·49-s − 0.840·51-s − 0.529·57-s − 0.128·61-s + 0.125·63-s − 1.71·67-s + 0.722·69-s + 0.712·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| 1171 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−13.26615901699558, −12.88366522341918, −12.11736335968340, −11.82337617607769, −11.08906014724698, −10.84527911007946, −10.35698796891444, −9.666522974956752, −9.378032621810917, −8.819997699792599, −8.409669218148813, −7.965772677847919, −7.684279115786893, −7.116887447860740, −6.399442256624316, −6.065966310829348, −5.311024511840141, −4.896641509000207, −4.249997185170188, −3.840662324823298, −3.104723478244310, −2.765327420538986, −2.109782500338846, −1.784262407089492, −0.8232229060231488, 0,
0.8232229060231488, 1.784262407089492, 2.109782500338846, 2.765327420538986, 3.104723478244310, 3.840662324823298, 4.249997185170188, 4.896641509000207, 5.311024511840141, 6.065966310829348, 6.399442256624316, 7.116887447860740, 7.684279115786893, 7.965772677847919, 8.409669218148813, 8.819997699792599, 9.378032621810917, 9.666522974956752, 10.35698796891444, 10.84527911007946, 11.08906014724698, 11.82337617607769, 12.11736335968340, 12.88366522341918, 13.26615901699558
