| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 6·6-s + 7·7-s + 8·8-s + 9·9-s + 28·11-s − 12·12-s − 54·13-s + 14·14-s + 16·16-s + 46·17-s + 18·18-s + 12·19-s − 21·21-s + 56·22-s − 24·24-s − 108·26-s − 27·27-s + 28·28-s + 6·29-s + 296·31-s + 32·32-s − 84·33-s + 92·34-s + 36·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.767·11-s − 0.288·12-s − 1.15·13-s + 0.267·14-s + 1/4·16-s + 0.656·17-s + 0.235·18-s + 0.144·19-s − 0.218·21-s + 0.542·22-s − 0.204·24-s − 0.814·26-s − 0.192·27-s + 0.188·28-s + 0.0384·29-s + 1.71·31-s + 0.176·32-s − 0.443·33-s + 0.464·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.989438188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.989438188\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 11 | \( 1 - 28 T + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 - 46 T + p^{3} T^{2} \) |
| 19 | \( 1 - 12 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 - 6 T + p^{3} T^{2} \) |
| 31 | \( 1 - 296 T + p^{3} T^{2} \) |
| 37 | \( 1 + 134 T + p^{3} T^{2} \) |
| 41 | \( 1 - 146 T + p^{3} T^{2} \) |
| 43 | \( 1 + 556 T + p^{3} T^{2} \) |
| 47 | \( 1 - 448 T + p^{3} T^{2} \) |
| 53 | \( 1 + 46 T + p^{3} T^{2} \) |
| 59 | \( 1 - 748 T + p^{3} T^{2} \) |
| 61 | \( 1 + 50 T + p^{3} T^{2} \) |
| 67 | \( 1 - 156 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1024 T + p^{3} T^{2} \) |
| 73 | \( 1 - 310 T + p^{3} T^{2} \) |
| 79 | \( 1 - 856 T + p^{3} T^{2} \) |
| 83 | \( 1 - 628 T + p^{3} T^{2} \) |
| 89 | \( 1 + 590 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1390 T + p^{3} T^{2} \) |
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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
−9.824542798628185480793866332228, −8.649871097169364945201489988316, −7.62018499647519722242540314372, −6.88356999050361749732154319361, −6.03675419126652532348006347097, −5.10966841476255349115981685237, −4.47158392869430093766008630833, −3.36553031196098204536733910677, −2.12736238151186049743952241399, −0.866393015044349164697381708036,
0.866393015044349164697381708036, 2.12736238151186049743952241399, 3.36553031196098204536733910677, 4.47158392869430093766008630833, 5.10966841476255349115981685237, 6.03675419126652532348006347097, 6.88356999050361749732154319361, 7.62018499647519722242540314372, 8.649871097169364945201489988316, 9.824542798628185480793866332228
