L(s) = 1 | − 2·3-s + 3·9-s + 2·11-s + 12·17-s − 6·25-s − 4·27-s − 4·33-s + 12·41-s − 16·43-s − 14·49-s − 24·51-s − 24·59-s + 8·67-s − 28·73-s + 12·75-s + 5·81-s − 24·83-s + 20·89-s − 28·97-s + 6·99-s + 8·107-s − 12·113-s + 3·121-s − 24·123-s + 127-s + 32·129-s + 131-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 9-s + 0.603·11-s + 2.91·17-s − 6/5·25-s − 0.769·27-s − 0.696·33-s + 1.87·41-s − 2.43·43-s − 2·49-s − 3.36·51-s − 3.12·59-s + 0.977·67-s − 3.27·73-s + 1.38·75-s + 5/9·81-s − 2.63·83-s + 2.11·89-s − 2.84·97-s + 0.603·99-s + 0.773·107-s − 1.12·113-s + 3/11·121-s − 2.16·123-s + 0.0887·127-s + 2.81·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 557568 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 557568 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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
−8.065545776521386769835087972572, −7.66759499524938235498389727755, −7.43093179715245875851298902341, −6.78625245992293517864906809126, −6.23040317898142022119405631367, −5.88581426070419577004157985232, −5.61117166658773050964433981813, −4.97830796048395948436336588916, −4.56394968516001415459865631360, −3.89899929611869403979836043364, −3.36387077635425811672598608724, −2.87113265788866454300212080099, −1.49507921150573825532250127010, −1.38445194475188199481237572528, 0,
1.38445194475188199481237572528, 1.49507921150573825532250127010, 2.87113265788866454300212080099, 3.36387077635425811672598608724, 3.89899929611869403979836043364, 4.56394968516001415459865631360, 4.97830796048395948436336588916, 5.61117166658773050964433981813, 5.88581426070419577004157985232, 6.23040317898142022119405631367, 6.78625245992293517864906809126, 7.43093179715245875851298902341, 7.66759499524938235498389727755, 8.065545776521386769835087972572