L(s) = 1 | + 2·3-s + 5-s + 7-s + 9-s − 11-s + 2·15-s + 2·21-s + 4·23-s + 25-s − 4·27-s + 2·29-s + 2·31-s − 2·33-s + 35-s − 6·37-s + 8·41-s + 12·43-s + 45-s + 6·47-s + 49-s − 6·53-s − 55-s + 10·59-s − 4·61-s + 63-s + 8·67-s + 8·69-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.516·15-s + 0.436·21-s + 0.834·23-s + 1/5·25-s − 0.769·27-s + 0.371·29-s + 0.359·31-s − 0.348·33-s + 0.169·35-s − 0.986·37-s + 1.24·41-s + 1.82·43-s + 0.149·45-s + 0.875·47-s + 1/7·49-s − 0.824·53-s − 0.134·55-s + 1.30·59-s − 0.512·61-s + 0.125·63-s + 0.977·67-s + 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.550333811\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.550333811\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.050211902943267076784687279670, −7.56267747122931163077444789139, −6.78056032363685606232921096912, −5.87997738209408176395348485438, −5.18933435514271702317430389940, −4.33125394928057241468119174674, −3.49279777040919907040866877211, −2.65605437910558752350573792184, −2.12726925429771010968869240672, −0.951657342869011093483118392059,
0.951657342869011093483118392059, 2.12726925429771010968869240672, 2.65605437910558752350573792184, 3.49279777040919907040866877211, 4.33125394928057241468119174674, 5.18933435514271702317430389940, 5.87997738209408176395348485438, 6.78056032363685606232921096912, 7.56267747122931163077444789139, 8.050211902943267076784687279670