| L(s) = 1 | + 3-s − 2·5-s + 9-s − 2·11-s + 4·13-s − 2·15-s − 6·17-s + 8·19-s + 6·23-s − 25-s + 27-s − 10·29-s + 4·31-s − 2·33-s + 6·37-s + 4·39-s + 6·41-s − 4·43-s − 2·45-s + 8·47-s − 6·51-s + 2·53-s + 4·55-s + 8·57-s − 4·59-s + 8·61-s − 8·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.603·11-s + 1.10·13-s − 0.516·15-s − 1.45·17-s + 1.83·19-s + 1.25·23-s − 1/5·25-s + 0.192·27-s − 1.85·29-s + 0.718·31-s − 0.348·33-s + 0.986·37-s + 0.640·39-s + 0.937·41-s − 0.609·43-s − 0.298·45-s + 1.16·47-s − 0.840·51-s + 0.274·53-s + 0.539·55-s + 1.05·57-s − 0.520·59-s + 1.02·61-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.859834953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.859834953\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−9.021448076657291772471054664206, −8.121630311281896118298701198515, −7.58175449833833984923350960350, −6.90110495522738730947232099166, −5.83476577963631855918351720949, −4.90650825242746693353421009887, −3.96167110552708099117817122276, −3.32094006681496489631591383434, −2.29815132805466510135803508307, −0.865485861345447747052784794456,
0.865485861345447747052784794456, 2.29815132805466510135803508307, 3.32094006681496489631591383434, 3.96167110552708099117817122276, 4.90650825242746693353421009887, 5.83476577963631855918351720949, 6.90110495522738730947232099166, 7.58175449833833984923350960350, 8.121630311281896118298701198515, 9.021448076657291772471054664206
