| L(s) = 1 | + 2-s + 4-s + 3·5-s + 7-s + 8-s + 3·10-s + 3·11-s − 4·13-s + 14-s + 16-s − 4·19-s + 3·20-s + 3·22-s + 6·23-s + 4·25-s − 4·26-s + 28-s + 3·29-s − 31-s + 32-s + 3·35-s + 2·37-s − 4·38-s + 3·40-s + 2·43-s + 3·44-s + 6·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.377·7-s + 0.353·8-s + 0.948·10-s + 0.904·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.917·19-s + 0.670·20-s + 0.639·22-s + 1.25·23-s + 4/5·25-s − 0.784·26-s + 0.188·28-s + 0.557·29-s − 0.179·31-s + 0.176·32-s + 0.507·35-s + 0.328·37-s − 0.648·38-s + 0.474·40-s + 0.304·43-s + 0.452·44-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.065490712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.065490712\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−14.58538064772913, −14.44027844382591, −13.99957202456193, −13.19961393121339, −13.02129211277252, −12.32663766237826, −11.92225339056163, −11.15003963958303, −10.80044067961548, −10.10604706339201, −9.585316063393907, −9.184578407102267, −8.495501897189264, −7.817053035026457, −7.028313543244097, −6.595637368104644, −6.178758483733613, −5.332392841189425, −5.057927583656458, −4.377017348548711, −3.690553956514740, −2.777394012319525, −2.271084656884187, −1.668778045954422, −0.8188225848263012,
0.8188225848263012, 1.668778045954422, 2.271084656884187, 2.777394012319525, 3.690553956514740, 4.377017348548711, 5.057927583656458, 5.332392841189425, 6.178758483733613, 6.595637368104644, 7.028313543244097, 7.817053035026457, 8.495501897189264, 9.184578407102267, 9.585316063393907, 10.10604706339201, 10.80044067961548, 11.15003963958303, 11.92225339056163, 12.32663766237826, 13.02129211277252, 13.19961393121339, 13.99957202456193, 14.44027844382591, 14.58538064772913
