| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 2·11-s − 14-s + 16-s + 2·17-s + 6·19-s − 20-s − 2·22-s + 6·23-s + 25-s − 28-s − 3·29-s + 5·31-s + 32-s + 2·34-s + 35-s + 5·37-s + 6·38-s − 40-s + 5·41-s + 4·43-s − 2·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.603·11-s − 0.267·14-s + 1/4·16-s + 0.485·17-s + 1.37·19-s − 0.223·20-s − 0.426·22-s + 1.25·23-s + 1/5·25-s − 0.188·28-s − 0.557·29-s + 0.898·31-s + 0.176·32-s + 0.342·34-s + 0.169·35-s + 0.821·37-s + 0.973·38-s − 0.158·40-s + 0.780·41-s + 0.609·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.486455576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.486455576\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.327310977038512413322641899790, −8.256960138680187788301884097654, −7.50159938372366565667360300009, −6.91149310511519256233376310838, −5.83534347977240772628407947692, −5.19860134238623823650204582017, −4.28024321587473415564217161482, −3.29552991695685832606529417876, −2.63059181977209316088438786145, −0.991473717787980877746384283119,
0.991473717787980877746384283119, 2.63059181977209316088438786145, 3.29552991695685832606529417876, 4.28024321587473415564217161482, 5.19860134238623823650204582017, 5.83534347977240772628407947692, 6.91149310511519256233376310838, 7.50159938372366565667360300009, 8.256960138680187788301884097654, 9.327310977038512413322641899790
