| L(s) = 1 | + 3-s − 7-s + 9-s + 2·13-s − 6·17-s − 4·19-s − 21-s − 4·23-s + 27-s + 6·29-s − 6·37-s + 2·39-s + 6·41-s + 4·43-s − 8·47-s + 49-s − 6·51-s − 14·53-s − 4·57-s − 4·59-s − 2·61-s − 63-s − 12·67-s − 4·69-s − 12·71-s − 10·73-s + 8·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.554·13-s − 1.45·17-s − 0.917·19-s − 0.218·21-s − 0.834·23-s + 0.192·27-s + 1.11·29-s − 0.986·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.840·51-s − 1.92·53-s − 0.529·57-s − 0.520·59-s − 0.256·61-s − 0.125·63-s − 1.46·67-s − 0.481·69-s − 1.42·71-s − 1.17·73-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−8.115476668361922334532340848855, −7.39777391959832010974227789838, −6.35146521903197022217050958088, −6.22822900029384084341990572645, −4.80355293522554958261058915364, −4.24599701108298073847497244207, −3.34772538059021110038642684561, −2.48248313295940449527693598752, −1.58753299698924332048386453958, 0,
1.58753299698924332048386453958, 2.48248313295940449527693598752, 3.34772538059021110038642684561, 4.24599701108298073847497244207, 4.80355293522554958261058915364, 6.22822900029384084341990572645, 6.35146521903197022217050958088, 7.39777391959832010974227789838, 8.115476668361922334532340848855
