| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s + 16-s − 2·19-s − 20-s − 2·23-s + 25-s + 32-s − 2·38-s − 40-s − 2·46-s + 2·47-s − 49-s + 50-s + 2·53-s + 64-s − 2·76-s − 80-s − 2·92-s + 2·94-s + 2·95-s − 98-s + 100-s + 2·106-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s + 16-s − 2·19-s − 20-s − 2·23-s + 25-s + 32-s − 2·38-s − 40-s − 2·46-s + 2·47-s − 49-s + 50-s + 2·53-s + 64-s − 2·76-s − 80-s − 2·92-s + 2·94-s + 2·95-s − 98-s + 100-s + 2·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.235538541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.235538541\) |
| \(L(1)\) |
|
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 | \( 1 + T \) |
| good | 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 + T )^{2} \) |
| 23 | \( ( 1 + T )^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )^{2} \) |
| 53 | \( ( 1 - T )^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−11.91167961398113772989984252920, −10.93573068856257254613494833130, −10.22931525894080399296453244859, −8.605097060411991331183090925000, −7.78848509001662843602149382606, −6.76744553658323464012569409468, −5.81132855144172084404809208196, −4.42594409490818214548157086073, −3.81630103763543940224861149259, −2.29727310763764692709638735892,
2.29727310763764692709638735892, 3.81630103763543940224861149259, 4.42594409490818214548157086073, 5.81132855144172084404809208196, 6.76744553658323464012569409468, 7.78848509001662843602149382606, 8.605097060411991331183090925000, 10.22931525894080399296453244859, 10.93573068856257254613494833130, 11.91167961398113772989984252920
