| L(s) = 1 | + 3-s + 4-s − 2·9-s + 12-s + 6·13-s + 16-s − 6·25-s − 5·27-s − 2·36-s + 6·39-s + 8·43-s + 48-s − 49-s + 6·52-s + 26·61-s + 64-s − 6·75-s + 30·79-s + 81-s − 6·100-s + 8·103-s − 5·108-s − 12·117-s + 3·121-s + 127-s + 8·129-s + 131-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s − 2/3·9-s + 0.288·12-s + 1.66·13-s + 1/4·16-s − 6/5·25-s − 0.962·27-s − 1/3·36-s + 0.960·39-s + 1.21·43-s + 0.144·48-s − 1/7·49-s + 0.832·52-s + 3.32·61-s + 1/8·64-s − 0.692·75-s + 3.37·79-s + 1/9·81-s − 3/5·100-s + 0.788·103-s − 0.481·108-s − 1.10·117-s + 3/11·121-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.491010045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.491010045\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−8.706637042837452115646683784524, −8.405143769012018590320970692875, −7.908098830137742952164634396654, −7.64991708074896242128992477533, −6.90080038208987052095905437235, −6.44017653769849185573375550807, −6.02026557348967301730278720270, −5.55873953403258015768087761302, −5.07969118487767146571584768668, −4.07449964933746898708074682402, −3.75080278252034506173191309361, −3.28015462412089861573993047608, −2.43615024430215349732491711883, −1.99519218986346290811935023896, −0.919220546883756431652662916661,
0.919220546883756431652662916661, 1.99519218986346290811935023896, 2.43615024430215349732491711883, 3.28015462412089861573993047608, 3.75080278252034506173191309361, 4.07449964933746898708074682402, 5.07969118487767146571584768668, 5.55873953403258015768087761302, 6.02026557348967301730278720270, 6.44017653769849185573375550807, 6.90080038208987052095905437235, 7.64991708074896242128992477533, 7.908098830137742952164634396654, 8.405143769012018590320970692875, 8.706637042837452115646683784524
