| L(s) = 1 | + 3-s − 7-s + 9-s − 13-s + 3·19-s − 21-s − 4·23-s + 27-s − 4·29-s + 7·31-s + 6·37-s − 39-s + 6·41-s + 9·43-s − 6·47-s − 6·49-s − 2·53-s + 3·57-s + 10·59-s + 61-s − 63-s − 3·67-s − 4·69-s + 14·71-s + 10·73-s − 8·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.277·13-s + 0.688·19-s − 0.218·21-s − 0.834·23-s + 0.192·27-s − 0.742·29-s + 1.25·31-s + 0.986·37-s − 0.160·39-s + 0.937·41-s + 1.37·43-s − 0.875·47-s − 6/7·49-s − 0.274·53-s + 0.397·57-s + 1.30·59-s + 0.128·61-s − 0.125·63-s − 0.366·67-s − 0.481·69-s + 1.66·71-s + 1.17·73-s − 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.267826177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.267826177\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.044507458992700649002783830433, −7.81354208028598196631112948002, −6.85384051528229043102893810217, −6.19043783667199575001617830610, −5.35678267081370894311957014372, −4.44428870987053500443935451372, −3.70977564500844432172292527712, −2.85301639523686726351590653315, −2.08063189124295169096705996485, −0.806219098277824264438793069433,
0.806219098277824264438793069433, 2.08063189124295169096705996485, 2.85301639523686726351590653315, 3.70977564500844432172292527712, 4.44428870987053500443935451372, 5.35678267081370894311957014372, 6.19043783667199575001617830610, 6.85384051528229043102893810217, 7.81354208028598196631112948002, 8.044507458992700649002783830433
