| L(s) = 1 | + 2·3-s − 3·4-s − 2·5-s + 3·9-s − 6·12-s − 4·15-s + 5·16-s + 6·20-s + 8·23-s + 3·25-s + 4·27-s + 16·31-s − 9·36-s − 12·37-s − 6·45-s + 16·47-s + 10·48-s − 10·49-s + 12·53-s + 12·60-s − 3·64-s + 24·67-s + 16·69-s + 16·71-s + 6·75-s − 10·80-s + 5·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 3/2·4-s − 0.894·5-s + 9-s − 1.73·12-s − 1.03·15-s + 5/4·16-s + 1.34·20-s + 1.66·23-s + 3/5·25-s + 0.769·27-s + 2.87·31-s − 3/2·36-s − 1.97·37-s − 0.894·45-s + 2.33·47-s + 1.44·48-s − 1.42·49-s + 1.64·53-s + 1.54·60-s − 3/8·64-s + 2.93·67-s + 1.92·69-s + 1.89·71-s + 0.692·75-s − 1.11·80-s + 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.382360294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.382360294\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.69344890685142555942945577499, −7.12224324890981759228654811108, −6.81358583409942750681700792930, −6.53038417539122512528573963592, −5.65462395656023150861273258602, −5.12106858409169630942967149663, −4.97190009229779476100580095986, −4.31338610647448896952875925848, −4.18139470284158080833278340409, −3.42381310446072218205035911280, −3.40996389133644129274747060882, −2.64207358991310319726851351291, −2.15956392843062891904606784720, −1.01147500994564153503812445251, −0.71020369206114523558950625222,
0.71020369206114523558950625222, 1.01147500994564153503812445251, 2.15956392843062891904606784720, 2.64207358991310319726851351291, 3.40996389133644129274747060882, 3.42381310446072218205035911280, 4.18139470284158080833278340409, 4.31338610647448896952875925848, 4.97190009229779476100580095986, 5.12106858409169630942967149663, 5.65462395656023150861273258602, 6.53038417539122512528573963592, 6.81358583409942750681700792930, 7.12224324890981759228654811108, 7.69344890685142555942945577499
