| L(s) = 1 | − 2·3-s + 3·9-s + 8·11-s − 6·25-s − 4·27-s − 16·33-s + 6·37-s − 12·41-s − 14·49-s − 4·53-s − 8·67-s + 16·71-s + 20·73-s + 12·75-s + 5·81-s − 8·83-s + 24·99-s − 36·101-s − 24·107-s − 12·111-s + 26·121-s + 24·123-s + 127-s + 131-s + 137-s + 139-s + 28·147-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 2.41·11-s − 6/5·25-s − 0.769·27-s − 2.78·33-s + 0.986·37-s − 1.87·41-s − 2·49-s − 0.549·53-s − 0.977·67-s + 1.89·71-s + 2.34·73-s + 1.38·75-s + 5/9·81-s − 0.878·83-s + 2.41·99-s − 3.58·101-s − 2.32·107-s − 1.13·111-s + 2.36·121-s + 2.16·123-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.30·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 788544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 788544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 37 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.098990694093691505068092710868, −7.54411782336909638415069910544, −6.79403995929460668221691676024, −6.60543423459238365209309705520, −6.42897107072744719896577758454, −5.85488214908078535460248247147, −5.27978213194301412304769875228, −4.89877774494537037497371281134, −4.25303028692796488061253338187, −3.82226398380683996966467971676, −3.52728755983219693796749664573, −2.50705776336759033940375936024, −1.53887243180198313451205963458, −1.29419434611716002821117874370, 0,
1.29419434611716002821117874370, 1.53887243180198313451205963458, 2.50705776336759033940375936024, 3.52728755983219693796749664573, 3.82226398380683996966467971676, 4.25303028692796488061253338187, 4.89877774494537037497371281134, 5.27978213194301412304769875228, 5.85488214908078535460248247147, 6.42897107072744719896577758454, 6.60543423459238365209309705520, 6.79403995929460668221691676024, 7.54411782336909638415069910544, 8.098990694093691505068092710868
