| L(s) = 1 | + 2·3-s + 4·5-s − 7-s + 9-s − 4·11-s + 4·13-s + 8·15-s − 2·17-s + 6·19-s − 2·21-s + 8·23-s + 11·25-s − 4·27-s + 6·29-s − 4·31-s − 8·33-s − 4·35-s + 2·37-s + 8·39-s − 41-s + 4·45-s − 4·47-s + 49-s − 4·51-s + 2·53-s − 16·55-s + 12·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.10·13-s + 2.06·15-s − 0.485·17-s + 1.37·19-s − 0.436·21-s + 1.66·23-s + 11/5·25-s − 0.769·27-s + 1.11·29-s − 0.718·31-s − 1.39·33-s − 0.676·35-s + 0.328·37-s + 1.28·39-s − 0.156·41-s + 0.596·45-s − 0.583·47-s + 1/7·49-s − 0.560·51-s + 0.274·53-s − 2.15·55-s + 1.58·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.145968280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.145968280\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.552213544158696581842047579887, −7.67416917162460708373949624115, −6.88549096455857967337624361946, −6.07245722278469324027916425920, −5.45867751595177148006868940602, −4.74625520105901773637245675588, −3.26372540237203182700292816004, −2.93978222550243599008946488915, −2.11673966692045370896465754544, −1.15180360808610940893880610998,
1.15180360808610940893880610998, 2.11673966692045370896465754544, 2.93978222550243599008946488915, 3.26372540237203182700292816004, 4.74625520105901773637245675588, 5.45867751595177148006868940602, 6.07245722278469324027916425920, 6.88549096455857967337624361946, 7.67416917162460708373949624115, 8.552213544158696581842047579887
