L(s) = 1 | + 3-s − 5-s − 2·9-s + 2·11-s − 15-s + 4·17-s − 2·19-s + 23-s + 25-s − 5·27-s + 9·29-s + 4·31-s + 2·33-s + 4·37-s + 41-s + 9·43-s + 2·45-s + 4·51-s − 10·53-s − 2·55-s − 2·57-s − 10·59-s + 9·61-s + 5·67-s + 69-s + 14·71-s + 12·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 2/3·9-s + 0.603·11-s − 0.258·15-s + 0.970·17-s − 0.458·19-s + 0.208·23-s + 1/5·25-s − 0.962·27-s + 1.67·29-s + 0.718·31-s + 0.348·33-s + 0.657·37-s + 0.156·41-s + 1.37·43-s + 0.298·45-s + 0.560·51-s − 1.37·53-s − 0.269·55-s − 0.264·57-s − 1.30·59-s + 1.15·61-s + 0.610·67-s + 0.120·69-s + 1.66·71-s + 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.952924545\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.952924545\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−9.184485842190697442035340982354, −8.149986311078947859120351383321, −8.018804392661812596144037671016, −6.80482359398900801205758108862, −6.11106241557641029420718429154, −5.07579799720258571099266026783, −4.11780576062108303639104457828, −3.25801088170597126170419177036, −2.43215850414134174046808592129, −0.931782521819548102167236757575,
0.931782521819548102167236757575, 2.43215850414134174046808592129, 3.25801088170597126170419177036, 4.11780576062108303639104457828, 5.07579799720258571099266026783, 6.11106241557641029420718429154, 6.80482359398900801205758108862, 8.018804392661812596144037671016, 8.149986311078947859120351383321, 9.184485842190697442035340982354