| L(s) = 1 | − 3.36·3-s − 1.08·7-s + 8.29·9-s − 11-s + 4·13-s − 0.107·17-s + 6.61·19-s + 3.63·21-s + 5.97·23-s − 17.7·27-s + 7.80·29-s − 1.12·31-s + 3.36·33-s − 7.05·37-s − 13.4·39-s + 5.19·41-s − 5.52·43-s + 7.69·47-s − 5.82·49-s + 0.362·51-s − 4.77·53-s − 22.2·57-s + 0.677·59-s + 0.197·61-s − 8.97·63-s + 1.41·67-s − 20.0·69-s + ⋯ |
| L(s) = 1 | − 1.93·3-s − 0.409·7-s + 2.76·9-s − 0.301·11-s + 1.10·13-s − 0.0261·17-s + 1.51·19-s + 0.793·21-s + 1.24·23-s − 3.42·27-s + 1.44·29-s − 0.202·31-s + 0.584·33-s − 1.15·37-s − 2.15·39-s + 0.810·41-s − 0.843·43-s + 1.12·47-s − 0.832·49-s + 0.0507·51-s − 0.656·53-s − 2.94·57-s + 0.0882·59-s + 0.0252·61-s − 1.13·63-s + 0.173·67-s − 2.41·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9994862333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9994862333\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 3.36T + 3T^{2} \) |
| 7 | \( 1 + 1.08T + 7T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 0.107T + 17T^{2} \) |
| 19 | \( 1 - 6.61T + 19T^{2} \) |
| 23 | \( 1 - 5.97T + 23T^{2} \) |
| 29 | \( 1 - 7.80T + 29T^{2} \) |
| 31 | \( 1 + 1.12T + 31T^{2} \) |
| 37 | \( 1 + 7.05T + 37T^{2} \) |
| 41 | \( 1 - 5.19T + 41T^{2} \) |
| 43 | \( 1 + 5.52T + 43T^{2} \) |
| 47 | \( 1 - 7.69T + 47T^{2} \) |
| 53 | \( 1 + 4.77T + 53T^{2} \) |
| 59 | \( 1 - 0.677T + 59T^{2} \) |
| 61 | \( 1 - 0.197T + 61T^{2} \) |
| 67 | \( 1 - 1.41T + 67T^{2} \) |
| 71 | \( 1 - 6.15T + 71T^{2} \) |
| 73 | \( 1 - 6.16T + 73T^{2} \) |
| 79 | \( 1 - 9.35T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 1.29T + 89T^{2} \) |
| 97 | \( 1 + 6.60T + 97T^{2} \) |
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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
−8.268413615864897203028985556662, −7.29803426301826336387366221553, −6.74589729309607577338250603592, −6.15903453667238992598234139391, −5.36350650248567106578935397097, −4.97403082922580755056050706263, −3.98129278211405132478120710227, −3.06492452418185620768315751241, −1.43586527728981454530773061443, −0.69166612315533654428956804481,
0.69166612315533654428956804481, 1.43586527728981454530773061443, 3.06492452418185620768315751241, 3.98129278211405132478120710227, 4.97403082922580755056050706263, 5.36350650248567106578935397097, 6.15903453667238992598234139391, 6.74589729309607577338250603592, 7.29803426301826336387366221553, 8.268413615864897203028985556662
