| L(s) = 1 | + 3-s − 3·4-s + 9-s − 3·12-s + 4·13-s + 5·16-s + 8·19-s + 27-s − 3·36-s + 20·37-s + 4·39-s − 8·43-s + 5·48-s − 14·49-s − 12·52-s + 8·57-s − 4·61-s − 3·64-s − 24·67-s − 20·73-s − 24·76-s + 81-s − 4·97-s + 32·103-s − 3·108-s + 28·109-s + 20·111-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 3/2·4-s + 1/3·9-s − 0.866·12-s + 1.10·13-s + 5/4·16-s + 1.83·19-s + 0.192·27-s − 1/2·36-s + 3.28·37-s + 0.640·39-s − 1.21·43-s + 0.721·48-s − 2·49-s − 1.66·52-s + 1.05·57-s − 0.512·61-s − 3/8·64-s − 2.93·67-s − 2.34·73-s − 2.75·76-s + 1/9·81-s − 0.406·97-s + 3.15·103-s − 0.288·108-s + 2.68·109-s + 1.89·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16875 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16875 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.032626388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.032626388\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−11.19897971912574422599178738873, −10.04517050900761317536685250745, −9.965958896202437987484594622667, −9.327820441539330420804608673051, −8.843956595114680827452383410006, −8.455181921164028207665051683361, −7.68003844385765985083943135134, −7.46261028773657489832215471574, −6.20681526904542088835730053039, −5.86497653842965752491328535183, −4.79492945983632365647604814265, −4.52160444884874669934496061646, −3.55714635244582915245852508954, −3.01549784140686353642765162463, −1.27025970316177507740293768624,
1.27025970316177507740293768624, 3.01549784140686353642765162463, 3.55714635244582915245852508954, 4.52160444884874669934496061646, 4.79492945983632365647604814265, 5.86497653842965752491328535183, 6.20681526904542088835730053039, 7.46261028773657489832215471574, 7.68003844385765985083943135134, 8.455181921164028207665051683361, 8.843956595114680827452383410006, 9.327820441539330420804608673051, 9.965958896202437987484594622667, 10.04517050900761317536685250745, 11.19897971912574422599178738873
