| L(s) = 1 | − 2-s + 4-s + 7·7-s − 8-s − 7·14-s + 16-s − 3·23-s − 25-s + 7·28-s − 8·31-s − 32-s + 6·41-s + 3·46-s + 21·47-s + 25·49-s + 50-s − 7·56-s + 8·62-s + 64-s + 9·71-s − 14·73-s + 7·79-s − 6·82-s + 9·89-s − 3·92-s − 21·94-s − 2·97-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 2.64·7-s − 0.353·8-s − 1.87·14-s + 1/4·16-s − 0.625·23-s − 1/5·25-s + 1.32·28-s − 1.43·31-s − 0.176·32-s + 0.937·41-s + 0.442·46-s + 3.06·47-s + 25/7·49-s + 0.141·50-s − 0.935·56-s + 1.01·62-s + 1/8·64-s + 1.06·71-s − 1.63·73-s + 0.787·79-s − 0.662·82-s + 0.953·89-s − 0.312·92-s − 2.16·94-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.494734235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.494734235\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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
−9.479451111599742321277276629882, −8.989182079933497564451719780622, −8.668562100012677547674930032876, −8.074914592528725680875456320694, −7.67474727638043495978149614889, −7.46578880651830551531520852799, −6.79112141955428696664171792343, −5.83713126987886701487069113214, −5.56690816869047754599106252617, −4.90883155719702944393675288879, −4.30279513226741130121525740567, −3.76048788309758278302367936493, −2.47416655866624945495708535709, −1.95152499573645948426066327154, −1.14511050514001596954262140878,
1.14511050514001596954262140878, 1.95152499573645948426066327154, 2.47416655866624945495708535709, 3.76048788309758278302367936493, 4.30279513226741130121525740567, 4.90883155719702944393675288879, 5.56690816869047754599106252617, 5.83713126987886701487069113214, 6.79112141955428696664171792343, 7.46578880651830551531520852799, 7.67474727638043495978149614889, 8.074914592528725680875456320694, 8.668562100012677547674930032876, 8.989182079933497564451719780622, 9.479451111599742321277276629882
