| L(s) = 1 | + 2·3-s − 2·5-s + 4·7-s + 9-s − 6·11-s + 2·13-s − 4·15-s + 2·17-s − 6·19-s + 8·21-s + 4·23-s − 25-s − 4·27-s − 29-s − 6·31-s − 12·33-s − 8·35-s + 2·37-s + 4·39-s + 2·41-s + 10·43-s − 2·45-s − 2·47-s + 9·49-s + 4·51-s + 10·53-s + 12·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s − 1.80·11-s + 0.554·13-s − 1.03·15-s + 0.485·17-s − 1.37·19-s + 1.74·21-s + 0.834·23-s − 1/5·25-s − 0.769·27-s − 0.185·29-s − 1.07·31-s − 2.08·33-s − 1.35·35-s + 0.328·37-s + 0.640·39-s + 0.312·41-s + 1.52·43-s − 0.298·45-s − 0.291·47-s + 9/7·49-s + 0.560·51-s + 1.37·53-s + 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.323119098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.323119098\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.64913618416885992687121528483, −12.70905976606632363210822926316, −11.29976427066417668553165574203, −10.62296992663263176964060013632, −8.881475557679452489265470553636, −8.042165076629360306011085472227, −7.61346245740251327448911449380, −5.33274354782517866544461877227, −3.96489742001566272466291177721, −2.38848237143160909896496110452,
2.38848237143160909896496110452, 3.96489742001566272466291177721, 5.33274354782517866544461877227, 7.61346245740251327448911449380, 8.042165076629360306011085472227, 8.881475557679452489265470553636, 10.62296992663263176964060013632, 11.29976427066417668553165574203, 12.70905976606632363210822926316, 13.64913618416885992687121528483
