L(s) = 1 | + 2-s − 4-s + 2·7-s − 3·8-s + 2·11-s + 4·13-s + 2·14-s − 16-s + 2·17-s − 19-s + 2·22-s − 4·23-s + 4·26-s − 2·28-s − 4·29-s + 5·32-s + 2·34-s − 38-s + 10·43-s − 2·44-s − 4·46-s + 12·47-s − 3·49-s − 4·52-s − 2·53-s − 6·56-s − 4·58-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.755·7-s − 1.06·8-s + 0.603·11-s + 1.10·13-s + 0.534·14-s − 1/4·16-s + 0.485·17-s − 0.229·19-s + 0.426·22-s − 0.834·23-s + 0.784·26-s − 0.377·28-s − 0.742·29-s + 0.883·32-s + 0.342·34-s − 0.162·38-s + 1.52·43-s − 0.301·44-s − 0.589·46-s + 1.75·47-s − 3/7·49-s − 0.554·52-s − 0.274·53-s − 0.801·56-s − 0.525·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.631093867\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.631093867\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−8.389912449121720656996349867142, −7.77956430365409239549597843670, −6.77413124496812103826508710021, −5.85480954907153188639309970018, −5.55398578846005673326571337740, −4.43079872452472503907360111774, −4.01941339893062407930039307696, −3.21997602080213271616123074869, −1.99218859787686854616586424528, −0.860571459661015059037425690333,
0.860571459661015059037425690333, 1.99218859787686854616586424528, 3.21997602080213271616123074869, 4.01941339893062407930039307696, 4.43079872452472503907360111774, 5.55398578846005673326571337740, 5.85480954907153188639309970018, 6.77413124496812103826508710021, 7.77956430365409239549597843670, 8.389912449121720656996349867142