| L(s) = 1 | + 3-s − 4-s + 3·5-s − 12-s + 3·15-s − 3·20-s + 5·25-s − 27-s − 31-s + 2·37-s − 3·47-s + 49-s − 3·59-s − 3·60-s + 64-s − 67-s + 5·75-s − 81-s − 93-s + 97-s − 5·100-s + 103-s + 108-s + 2·111-s − 3·113-s + 124-s + 6·125-s + ⋯ |
| L(s) = 1 | + 3-s − 4-s + 3·5-s − 12-s + 3·15-s − 3·20-s + 5·25-s − 27-s − 31-s + 2·37-s − 3·47-s + 49-s − 3·59-s − 3·60-s + 64-s − 67-s + 5·75-s − 81-s − 93-s + 97-s − 5·100-s + 103-s + 108-s + 2·111-s − 3·113-s + 124-s + 6·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.807645711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.807645711\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
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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
−9.939132962556423288986642152861, −9.806273550964983182110806312035, −9.281402225724897303810430602015, −9.184596196807808871983297595491, −8.952908000561139503204809014889, −8.432524004762199201255146302930, −7.84286549193142732470097982057, −7.57441967524080583468446067987, −6.77533165825582423934110441318, −6.24625777462337605502399224789, −6.12336822320508778587964723993, −5.64355596691664070058038253679, −5.01898316164420595680900820225, −4.90793633165688835499495849715, −4.17134878020331786274540355984, −3.50145038384949733472795913469, −2.76577272010863711402446680769, −2.58106642770042651491156260282, −1.78050740651467284916510559322, −1.48666360492076691150571402513,
1.48666360492076691150571402513, 1.78050740651467284916510559322, 2.58106642770042651491156260282, 2.76577272010863711402446680769, 3.50145038384949733472795913469, 4.17134878020331786274540355984, 4.90793633165688835499495849715, 5.01898316164420595680900820225, 5.64355596691664070058038253679, 6.12336822320508778587964723993, 6.24625777462337605502399224789, 6.77533165825582423934110441318, 7.57441967524080583468446067987, 7.84286549193142732470097982057, 8.432524004762199201255146302930, 8.952908000561139503204809014889, 9.184596196807808871983297595491, 9.281402225724897303810430602015, 9.806273550964983182110806312035, 9.939132962556423288986642152861
