| L(s) = 1 | + 3·3-s − 6·5-s + 7·7-s + 9·9-s + 4·11-s + 46·13-s − 18·15-s − 82·17-s − 84·19-s + 21·21-s + 44·23-s − 89·25-s + 27·27-s − 70·29-s − 152·31-s + 12·33-s − 42·35-s + 146·37-s + 138·39-s + 94·41-s − 488·43-s − 54·45-s − 32·47-s + 49·49-s − 246·51-s + 562·53-s − 24·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.536·5-s + 0.377·7-s + 1/3·9-s + 0.109·11-s + 0.981·13-s − 0.309·15-s − 1.16·17-s − 1.01·19-s + 0.218·21-s + 0.398·23-s − 0.711·25-s + 0.192·27-s − 0.448·29-s − 0.880·31-s + 0.0633·33-s − 0.202·35-s + 0.648·37-s + 0.566·39-s + 0.358·41-s − 1.73·43-s − 0.178·45-s − 0.0993·47-s + 1/7·49-s − 0.675·51-s + 1.45·53-s − 0.0588·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 11 | \( 1 - 4 T + p^{3} T^{2} \) |
| 13 | \( 1 - 46 T + p^{3} T^{2} \) |
| 17 | \( 1 + 82 T + p^{3} T^{2} \) |
| 19 | \( 1 + 84 T + p^{3} T^{2} \) |
| 23 | \( 1 - 44 T + p^{3} T^{2} \) |
| 29 | \( 1 + 70 T + p^{3} T^{2} \) |
| 31 | \( 1 + 152 T + p^{3} T^{2} \) |
| 37 | \( 1 - 146 T + p^{3} T^{2} \) |
| 41 | \( 1 - 94 T + p^{3} T^{2} \) |
| 43 | \( 1 + 488 T + p^{3} T^{2} \) |
| 47 | \( 1 + 32 T + p^{3} T^{2} \) |
| 53 | \( 1 - 562 T + p^{3} T^{2} \) |
| 59 | \( 1 - 476 T + p^{3} T^{2} \) |
| 61 | \( 1 + 34 T + p^{3} T^{2} \) |
| 67 | \( 1 - 520 T + p^{3} T^{2} \) |
| 71 | \( 1 + 36 T + p^{3} T^{2} \) |
| 73 | \( 1 + 654 T + p^{3} T^{2} \) |
| 79 | \( 1 + 608 T + p^{3} T^{2} \) |
| 83 | \( 1 + 284 T + p^{3} T^{2} \) |
| 89 | \( 1 + 954 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1694 T + p^{3} T^{2} \) |
| 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
−8.619267736811403960976020295321, −8.276580887556976453892342959665, −7.24951359723626415762311874550, −6.51952174266162332844099314004, −5.45264282499444907361981540118, −4.25370339519307336900108911094, −3.78741033415156953704768457901, −2.50892418005340985278600030173, −1.49825965505686647185693823041, 0,
1.49825965505686647185693823041, 2.50892418005340985278600030173, 3.78741033415156953704768457901, 4.25370339519307336900108911094, 5.45264282499444907361981540118, 6.51952174266162332844099314004, 7.24951359723626415762311874550, 8.276580887556976453892342959665, 8.619267736811403960976020295321
